Precalculus Examples

$x {(1 - x^{2})}^{3} > 7 {(1 - x^{2})}^{3}$

Step 1

Simplify $7 {(1 - x^{2})}^{3}$ .

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Step 1.1

Use the Binomial Theorem.

$x {(1 - x^{2})}^{3} > 7 (1^{3} + 3 \cdot 1^{2} (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 1.2

Simplify terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

One to any power is one.

$x {(1 - x^{2})}^{3} > 7 (1 + 3 \cdot 1^{2} (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 1.2.1.2

One to any power is one.

$x {(1 - x^{2})}^{3} > 7 (1 + 3 \cdot 1 (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 1.2.1.3

Multiply $3$ by $1$ .

$x {(1 - x^{2})}^{3} > 7 (1 + 3 (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 1.2.1.4

Multiply $- 1$ by $3$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 1.2.1.5

Multiply $3$ by $1$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 1.2.1.6

Apply the product rule to $- x^{2}$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 ({(- 1)}^{2} {(x^{2})}^{2}) + {(- x^{2})}^{3})$

Step 1.2.1.7

Raise $- 1$ to the power of $2$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 (1 {(x^{2})}^{2}) + {(- x^{2})}^{3})$

Step 1.2.1.8

Multiply ${(x^{2})}^{2}$ by $1$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 {(x^{2})}^{2} + {(- x^{2})}^{3})$

Step 1.2.1.9

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.1.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 x^{2 \cdot 2} + {(- x^{2})}^{3})$

Step 1.2.1.9.2

Multiply $2$ by $2$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 x^{4} + {(- x^{2})}^{3})$

Step 1.2.1.10

Apply the product rule to $- x^{2}$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 x^{4} + {(- 1)}^{3} {(x^{2})}^{3})$

Step 1.2.1.11

Raise $- 1$ to the power of $3$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 x^{4} - {(x^{2})}^{3})$

Step 1.2.1.12

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 1.2.1.12.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 x^{4} - x^{2 \cdot 3})$

Step 1.2.1.12.2

Multiply $2$ by $3$ .

$x {(1 - x^{2})}^{3} > 7 (1 - 3 x^{2} + 3 x^{4} - x^{6})$

Step 1.2.2

Apply the distributive property.

$x {(1 - x^{2})}^{3} > 7 \cdot 1 + 7 (- 3 x^{2}) + 7 (3 x^{4}) + 7 (- x^{6})$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $7$ by $1$ .

$x {(1 - x^{2})}^{3} > 7 + 7 (- 3 x^{2}) + 7 (3 x^{4}) + 7 (- x^{6})$

Step 1.3.2

Multiply $- 3$ by $7$ .

$x {(1 - x^{2})}^{3} > 7 - 21 x^{2} + 7 (3 x^{4}) + 7 (- x^{6})$

Step 1.3.3

Multiply $3$ by $7$ .

$x {(1 - x^{2})}^{3} > 7 - 21 x^{2} + 21 x^{4} + 7 (- x^{6})$

Step 1.3.4

Multiply $- 1$ by $7$ .

$x {(1 - x^{2})}^{3} > 7 - 21 x^{2} + 21 x^{4} - 7 x^{6}$

Step 2

Rewrite so $x$ is on the left side of the inequality.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x {(1 - x^{2})}^{3}$

Step 3

Simplify $x {(1 - x^{2})}^{3}$ .

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Step 3.1

Use the Binomial Theorem.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1^{3} + 3 \cdot 1^{2} (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 3.2

Simplify terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

One to any power is one.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 + 3 \cdot 1^{2} (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 3.2.1.2

One to any power is one.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 + 3 \cdot 1 (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 3.2.1.3

Multiply $3$ by $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 + 3 (- x^{2}) + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 3.2.1.4

Multiply $- 1$ by $3$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 \cdot 1 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 3.2.1.5

Multiply $3$ by $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 {(- x^{2})}^{2} + {(- x^{2})}^{3})$

Step 3.2.1.6

Apply the product rule to $- x^{2}$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 ({(- 1)}^{2} {(x^{2})}^{2}) + {(- x^{2})}^{3})$

Step 3.2.1.7

Raise $- 1$ to the power of $2$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 (1 {(x^{2})}^{2}) + {(- x^{2})}^{3})$

Step 3.2.1.8

Multiply ${(x^{2})}^{2}$ by $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 {(x^{2})}^{2} + {(- x^{2})}^{3})$

Step 3.2.1.9

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.2.1.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 x^{2 \cdot 2} + {(- x^{2})}^{3})$

Step 3.2.1.9.2

Multiply $2$ by $2$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 x^{4} + {(- x^{2})}^{3})$

Step 3.2.1.10

Apply the product rule to $- x^{2}$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 x^{4} + {(- 1)}^{3} {(x^{2})}^{3})$

Step 3.2.1.11

Raise $- 1$ to the power of $3$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 x^{4} - {(x^{2})}^{3})$

Step 3.2.1.12

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 3.2.1.12.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 x^{4} - x^{2 \cdot 3})$

Step 3.2.1.12.2

Multiply $2$ by $3$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x (1 - 3 x^{2} + 3 x^{4} - x^{6})$

Step 3.2.2

Apply the distributive property.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x \cdot 1 + x (- 3 x^{2}) + x (3 x^{4}) + x (- x^{6})$

Step 3.3

Simplify.

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Step 3.3.1

Multiply $x$ by $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x + x (- 3 x^{2}) + x (3 x^{4}) + x (- x^{6})$

Step 3.3.2

Rewrite using the commutative property of multiplication.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x \cdot x^{2} + x (3 x^{4}) + x (- x^{6})$

Step 3.3.3

Rewrite using the commutative property of multiplication.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x \cdot x^{2} + 3 x \cdot x^{4} + x (- x^{6})$

Step 3.3.4

Rewrite using the commutative property of multiplication.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x \cdot x^{2} + 3 x \cdot x^{4} - x \cdot x^{6}$

Step 3.4

Simplify each term.

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Step 3.4.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.4.1.1

Move $x^{2}$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 (x^{2} x) + 3 x \cdot x^{4} - x \cdot x^{6}$

Step 3.4.1.2

Multiply $x^{2}$ by $x$ .

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Step 3.4.1.2.1

Raise $x$ to the power of $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 (x^{2} x^{1}) + 3 x \cdot x^{4} - x \cdot x^{6}$

Step 3.4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{2 + 1} + 3 x \cdot x^{4} - x \cdot x^{6}$

Step 3.4.1.3

Add $2$ and $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 x \cdot x^{4} - x \cdot x^{6}$

Step 3.4.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 3.4.2.1

Move $x^{4}$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 (x^{4} x) - x \cdot x^{6}$

Step 3.4.2.2

Multiply $x^{4}$ by $x$ .

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Step 3.4.2.2.1

Raise $x$ to the power of $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 (x^{4} x^{1}) - x \cdot x^{6}$

Step 3.4.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 x^{4 + 1} - x \cdot x^{6}$

Step 3.4.2.3

Add $4$ and $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 x^{5} - x \cdot x^{6}$

Step 3.4.3

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 3.4.3.1

Move $x^{6}$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 x^{5} - (x^{6} x)$

Step 3.4.3.2

Multiply $x^{6}$ by $x$ .

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Step 3.4.3.2.1

Raise $x$ to the power of $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 x^{5} - (x^{6} x^{1})$

Step 3.4.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 x^{5} - x^{6 + 1}$

Step 3.4.3.3

Add $6$ and $1$ .

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} < x - 3 x^{3} + 3 x^{5} - x^{7}$

Step 4

Move all terms containing $x$ to the left side of the inequality.

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Step 4.1

Subtract $x$ from both sides of the inequality.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} - x < - 3 x^{3} + 3 x^{5} - x^{7}$

Step 4.2

Add $3 x^{3}$ to both sides of the inequality.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} - x + 3 x^{3} < 3 x^{5} - x^{7}$

Step 4.3

Subtract $3 x^{5}$ from both sides of the inequality.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} - x + 3 x^{3} - 3 x^{5} < - x^{7}$

Step 4.4

Add $x^{7}$ to both sides of the inequality.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} - x + 3 x^{3} - 3 x^{5} + x^{7} < 0$

Step 5

Convert the inequality to an equation.

$7 - 21 x^{2} + 21 x^{4} - 7 x^{6} - x + 3 x^{3} - 3 x^{5} + x^{7} = 0$

Step 6

Factor the left side of the equation.

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Step 6.1

Reorder terms.

$x^{7} - 7 x^{6} - 3 x^{5} + 21 x^{4} + 3 x^{3} - 21 x^{2} - x + 7 = 0$

Step 6.2

Factor $x^{7} - 7 x^{6} - 3 x^{5} + 21 x^{4} + 3 x^{3} - 21 x^{2} - x + 7$ using the rational roots test.

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Step 6.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 7$

$q = \pm 1$

Step 6.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 7$

Step 6.2.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 6.2.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{7} - 7 {(- 1)}^{6} - 3 {(- 1)}^{5} + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.2

Raise $- 1$ to the power of $7$ .

$- 1 - 7 {(- 1)}^{6} - 3 {(- 1)}^{5} + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.3

Raise $- 1$ to the power of $6$ .

$- 1 - 7 \cdot 1 - 3 {(- 1)}^{5} + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.4

Multiply $- 7$ by $1$ .

$- 1 - 7 - 3 {(- 1)}^{5} + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.5

Subtract $7$ from $- 1$ .

$- 8 - 3 {(- 1)}^{5} + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.6

Raise $- 1$ to the power of $5$ .

$- 8 - 3 \cdot - 1 + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.7

Multiply $- 3$ by $- 1$ .

$- 8 + 3 + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.8

Add $- 8$ and $3$ .

$- 5 + 21 {(- 1)}^{4} + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.9

Raise $- 1$ to the power of $4$ .

$- 5 + 21 \cdot 1 + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.10

Multiply $21$ by $1$ .

$- 5 + 21 + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.11

Add $- 5$ and $21$ .

$16 + 3 {(- 1)}^{3} - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.12

Raise $- 1$ to the power of $3$ .

$16 + 3 \cdot - 1 - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.13

Multiply $3$ by $- 1$ .

$16 - 3 - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.14

Subtract $3$ from $16$ .

$13 - 21 {(- 1)}^{2} + 1 + 7$

Step 6.2.3.15

Raise $- 1$ to the power of $2$ .

$13 - 21 \cdot 1 + 1 + 7$

Step 6.2.3.16

Multiply $- 21$ by $1$ .

$13 - 21 + 1 + 7$

Step 6.2.3.17

Subtract $21$ from $13$ .

$- 8 + 1 + 7$

Step 6.2.3.18

Add $- 8$ and $1$ .

$- 7 + 7$

Step 6.2.3.19

Add $- 7$ and $7$ .

$0$

Step 6.2.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{7} - 7 x^{6} - 3 x^{5} + 21 x^{4} + 3 x^{3} - 21 x^{2} - x + 7}{x + 1}$

Step 6.2.5

Divide $x^{7} - 7 x^{6} - 3 x^{5} + 21 x^{4} + 3 x^{3} - 21 x^{2} - x + 7$ by $x + 1$ .

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Step 6.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

Step 6.2.5.2

Divide the highest order term in the dividend $x^{7}$ by the highest order term in divisor $x$ .

$x^{6}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

Step 6.2.5.3

Multiply the new quotient term by the divisor.

$x^{6}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

Step 6.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{7} + x^{6}$

$x^{6}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

Step 6.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{6}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

Step 6.2.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{6}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

Step 6.2.5.7

Divide the highest order term in the dividend $- 8 x^{6}$ by the highest order term in divisor $x$ .

$x^{6}$

$8 x^{5}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

Step 6.2.5.8

Multiply the new quotient term by the divisor.

$x^{6}$

$8 x^{5}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

Step 6.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{6} - 8 x^{5}$

$x^{6}$

$8 x^{5}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

Step 6.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{6}$

$8 x^{5}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

Step 6.2.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{6}$

$8 x^{5}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

Step 6.2.5.12

Divide the highest order term in the dividend $5 x^{5}$ by the highest order term in divisor $x$ .

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

Step 6.2.5.13

Multiply the new quotient term by the divisor.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

Step 6.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{5} + 5 x^{4}$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

Step 6.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

Step 6.2.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

Step 6.2.5.17

Divide the highest order term in the dividend $16 x^{4}$ by the highest order term in divisor $x$ .

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

Step 6.2.5.18

Multiply the new quotient term by the divisor.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

Step 6.2.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{4} + 16 x^{3}$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

Step 6.2.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

Step 6.2.5.21

Pull the next terms from the original dividend down into the current dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

Step 6.2.5.22

Divide the highest order term in the dividend $- 13 x^{3}$ by the highest order term in divisor $x$ .

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

Step 6.2.5.23

Multiply the new quotient term by the divisor.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

Step 6.2.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{3} - 13 x^{2}$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

Step 6.2.5.25

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

Step 6.2.5.26

Pull the next terms from the original dividend down into the current dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

Step 6.2.5.27

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

Step 6.2.5.28

Multiply the new quotient term by the divisor.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

Step 6.2.5.29

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 8 x$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

Step 6.2.5.30

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

$7 x$

Step 6.2.5.31

Pull the next terms from the original dividend down into the current dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

$7 x$

$7$

Step 6.2.5.32

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

$7 x$

$7$

Step 6.2.5.33

Multiply the new quotient term by the divisor.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

$7 x$

$7$

$7 x$

$7$

Step 6.2.5.34

The expression needs to be subtracted from the dividend, so change all the signs in $7 x + 7$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

$7 x$

$7$

$7 x$

$7$

Step 6.2.5.35

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x$

$1$

$x^{7}$

$7 x^{6}$

$3 x^{5}$

$21 x^{4}$

$3 x^{3}$

$21 x^{2}$

$x$

$7$

$x^{7}$

$x^{6}$

$8 x^{6}$

$3 x^{5}$

$8 x^{6}$

$8 x^{5}$

$5 x^{5}$

$21 x^{4}$

$5 x^{5}$

$5 x^{4}$

$16 x^{4}$

$3 x^{3}$

$16 x^{4}$

$16 x^{3}$

$13 x^{3}$

$21 x^{2}$

$13 x^{3}$

$13 x^{2}$

$8 x^{2}$

$x$

$8 x^{2}$

$8 x$

$7 x$

$7$

$7 x$

$7$

$0$

Step 6.2.5.36

Since the remander is $0$ , the final answer is the quotient.

$x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7$

Step 6.2.6

Write $x^{7} - 7 x^{6} - 3 x^{5} + 21 x^{4} + 3 x^{3} - 21 x^{2} - x + 7$ as a set of factors.

$(x + 1) (x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7) = 0$

Step 7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7 = 0$

Step 8

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9

Set $x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7$ equal to $0$ .

$x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7 = 0$

Step 9.2

Solve $x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7 = 0$ for $x$ .

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Step 9.2.1

Factor the left side of the equation.

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Step 9.2.1.1

Factor $x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7$ using the rational roots test.

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Step 9.2.1.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 7$

$q = \pm 1$

Step 9.2.1.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 7$

Step 9.2.1.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 9.2.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{6} - 8 {(- 1)}^{5} + 5 {(- 1)}^{4} + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.2

Raise $- 1$ to the power of $6$ .

$1 - 8 {(- 1)}^{5} + 5 {(- 1)}^{4} + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.3

Raise $- 1$ to the power of $5$ .

$1 - 8 \cdot - 1 + 5 {(- 1)}^{4} + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.4

Multiply $- 8$ by $- 1$ .

$1 + 8 + 5 {(- 1)}^{4} + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.5

Add $1$ and $8$ .

$9 + 5 {(- 1)}^{4} + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.6

Raise $- 1$ to the power of $4$ .

$9 + 5 \cdot 1 + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.7

Multiply $5$ by $1$ .

$9 + 5 + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.8

Add $9$ and $5$ .

$14 + 16 {(- 1)}^{3} - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.9

Raise $- 1$ to the power of $3$ .

$14 + 16 \cdot - 1 - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.10

Multiply $16$ by $- 1$ .

$14 - 16 - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.11

Subtract $16$ from $14$ .

$- 2 - 13 {(- 1)}^{2} - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.12

Raise $- 1$ to the power of $2$ .

$- 2 - 13 \cdot 1 - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.13

Multiply $- 13$ by $1$ .

$- 2 - 13 - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.14

Subtract $13$ from $- 2$ .

$- 15 - 8 \cdot - 1 + 7$

Step 9.2.1.1.3.15

Multiply $- 8$ by $- 1$ .

$- 15 + 8 + 7$

Step 9.2.1.1.3.16

Add $- 15$ and $8$ .

$- 7 + 7$

Step 9.2.1.1.3.17

Add $- 7$ and $7$ .

$0$

Step 9.2.1.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7}{x + 1}$

Step 9.2.1.1.5

Divide $x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7$ by $x + 1$ .

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Step 9.2.1.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

Step 9.2.1.1.5.2

Divide the highest order term in the dividend $x^{6}$ by the highest order term in divisor $x$ .

$x^{5}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

Step 9.2.1.1.5.3

Multiply the new quotient term by the divisor.

$x^{5}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

Step 9.2.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{6} + x^{5}$

$x^{5}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

Step 9.2.1.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

Step 9.2.1.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

Step 9.2.1.1.5.7

Divide the highest order term in the dividend $- 9 x^{5}$ by the highest order term in divisor $x$ .

$x^{5}$

$9 x^{4}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

Step 9.2.1.1.5.8

Multiply the new quotient term by the divisor.

$x^{5}$

$9 x^{4}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

Step 9.2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{5} - 9 x^{4}$

$x^{5}$

$9 x^{4}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

Step 9.2.1.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$9 x^{4}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

Step 9.2.1.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$9 x^{4}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

Step 9.2.1.1.5.12

Divide the highest order term in the dividend $14 x^{4}$ by the highest order term in divisor $x$ .

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

Step 9.2.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

Step 9.2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $14 x^{4} + 14 x^{3}$

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

Step 9.2.1.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

Step 9.2.1.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

Step 9.2.1.1.5.17

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

Step 9.2.1.1.5.18

Multiply the new quotient term by the divisor.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

Step 9.2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 2 x^{2}$

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

Step 9.2.1.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

Step 9.2.1.1.5.21

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

Step 9.2.1.1.5.22

Divide the highest order term in the dividend $- 15 x^{2}$ by the highest order term in divisor $x$ .

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

Step 9.2.1.1.5.23

Multiply the new quotient term by the divisor.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

Step 9.2.1.1.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{2} - 15 x$

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

Step 9.2.1.1.5.25

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

$7 x$

Step 9.2.1.1.5.26

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

$7 x$

$7$

Step 9.2.1.1.5.27

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$7$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

$7 x$

$7$

Step 9.2.1.1.5.28

Multiply the new quotient term by the divisor.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$7$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

$7 x$

$7$

$7 x$

$7$

Step 9.2.1.1.5.29

The expression needs to be subtracted from the dividend, so change all the signs in $7 x + 7$

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$7$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

$7 x$

$7$

$7 x$

$7$

Step 9.2.1.1.5.30

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$9 x^{4}$

$14 x^{3}$

$2 x^{2}$

$15 x$

$7$

$x$

$1$

$x^{6}$

$8 x^{5}$

$5 x^{4}$

$16 x^{3}$

$13 x^{2}$

$8 x$

$7$

$x^{6}$

$x^{5}$

$9 x^{5}$

$5 x^{4}$

$9 x^{5}$

$9 x^{4}$

$14 x^{4}$

$16 x^{3}$

$14 x^{4}$

$14 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$2 x^{2}$

$15 x^{2}$

$8 x$

$15 x^{2}$

$15 x$

$7 x$

$7$

$7 x$

$7$

$0$

Step 9.2.1.1.5.31

Since the remander is $0$ , the final answer is the quotient.

$x^{5} - 9 x^{4} + 14 x^{3} + 2 x^{2} - 15 x + 7$

Step 9.2.1.1.6

Write $x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7$ as a set of factors.

$(x + 1) (x^{5} - 9 x^{4} + 14 x^{3} + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.2

Factor $x^{3}$ out of $x^{5} - 9 x^{4} + 14 x^{3}$ .

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Step 9.2.1.2.1

Factor $x^{3}$ out of $x^{5}$ .

$(x + 1) (x^{3} x^{2} - 9 x^{4} + 14 x^{3} + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.2.2

Factor $x^{3}$ out of $- 9 x^{4}$ .

$(x + 1) (x^{3} x^{2} + x^{3} (- 9 x) + 14 x^{3} + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.2.3

Factor $x^{3}$ out of $14 x^{3}$ .

$(x + 1) (x^{3} x^{2} + x^{3} (- 9 x) + x^{3} \cdot 14 + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.2.4

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (- 9 x)$ .

$(x + 1) (x^{3} (x^{2} - 9 x) + x^{3} \cdot 14 + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.2.5

Factor $x^{3}$ out of $x^{3} (x^{2} - 9 x) + x^{3} \cdot 14$ .

$(x + 1) (x^{3} (x^{2} - 9 x + 14) + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.3

Factor.

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Step 9.2.1.3.1

Factor $x^{2} - 9 x + 14$ using the AC method.

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Step 9.2.1.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 9.2.1.3.1.2

Write the factored form using these integers.

$(x + 1) (x^{3} ((x - 7) (x - 2)) + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.3.2

Remove unnecessary parentheses.

$(x + 1) (x^{3} (x - 7) (x - 2) + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.4

Factor by grouping.

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Step 9.2.1.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 7 = 14$ and whose sum is $b = - 15$ .

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Step 9.2.1.4.1.1

Factor $- 15$ out of $- 15 x$ .

$(x + 1) (x^{3} (x - 7) (x - 2) + 2 x^{2} - 15 x + 7) = 0$

Step 9.2.1.4.1.2

Rewrite $- 15$ as $- 1$ plus $- 14$

$(x + 1) (x^{3} (x - 7) (x - 2) + 2 x^{2} + (- 1 - 14) x + 7) = 0$

Step 9.2.1.4.1.3

Apply the distributive property.

$(x + 1) (x^{3} (x - 7) (x - 2) + 2 x^{2} - 1 x - 14 x + 7) = 0$

Step 9.2.1.4.2

Factor out the greatest common factor from each group.

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Step 9.2.1.4.2.1

Group the first two terms and the last two terms.

$(x + 1) (x^{3} (x - 7) (x - 2) + 2 x^{2} - 1 x - 14 x + 7) = 0$

Step 9.2.1.4.2.2

Factor out the greatest common factor (GCF) from each group.

$(x + 1) (x^{3} (x - 7) (x - 2) + x (2 x - 1) - 7 (2 x - 1)) = 0$

Step 9.2.1.4.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(x + 1) (x^{3} (x - 7) (x - 2) + (2 x - 1) (x - 7)) = 0$

Step 9.2.1.5

Factor $x - 7$ out of $x^{3} (x - 7) (x - 2) + (2 x - 1) (x - 7)$ .

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Step 9.2.1.5.1

Factor $x - 7$ out of $x^{3} (x - 7) (x - 2)$ .

$(x + 1) ((x - 7) (x^{3} (x - 2)) + (2 x - 1) (x - 7)) = 0$

Step 9.2.1.5.2

Factor $x - 7$ out of $(2 x - 1) (x - 7)$ .

$(x + 1) ((x - 7) (x^{3} (x - 2)) + (x - 7) (2 x - 1)) = 0$

Step 9.2.1.5.3

Factor $x - 7$ out of $(x - 7) (x^{3} (x - 2)) + (x - 7) (2 x - 1)$ .

$(x + 1) ((x - 7) (x^{3} (x - 2) + 2 x - 1)) = 0$

Step 9.2.1.6

Apply the distributive property.

$(x + 1) ((x - 7) (x^{3} x + x^{3} \cdot - 2 + 2 x - 1)) = 0$

Step 9.2.1.7

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 9.2.1.7.1

Multiply $x^{3}$ by $x$ .

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Step 9.2.1.7.1.1

Raise $x$ to the power of $1$ .

$(x + 1) ((x - 7) (x^{3} x + x^{3} \cdot - 2 + 2 x - 1)) = 0$

Step 9.2.1.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 1) ((x - 7) (x^{3 + 1} + x^{3} \cdot - 2 + 2 x - 1)) = 0$

Step 9.2.1.7.2

Add $3$ and $1$ .

$(x + 1) ((x - 7) (x^{4} + x^{3} \cdot - 2 + 2 x - 1)) = 0$

Step 9.2.1.8

Move $- 2$ to the left of $x^{3}$ .

$(x + 1) ((x - 7) (x^{4} - 2 x^{3} + 2 x - 1)) = 0$

Step 9.2.1.9

Factor.

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Step 9.2.1.9.1

Factor.

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Step 9.2.1.9.1.1

Rewrite $x^{4} - 2 x^{3} + 2 x - 1$ in a factored form.

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Step 9.2.1.9.1.1.1

Factor $x^{4} - 2 x^{3} + 2 x - 1$ using the rational roots test.

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Step 9.2.1.9.1.1.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1$

Step 9.2.1.9.1.1.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1$

Step 9.2.1.9.1.1.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 9.2.1.9.1.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 2 {(- 1)}^{3} + 2 \cdot - 1 - 1$

Step 9.2.1.9.1.1.1.3.2

Raise $- 1$ to the power of $4$ .

$1 - 2 {(- 1)}^{3} + 2 \cdot - 1 - 1$

Step 9.2.1.9.1.1.1.3.3

Raise $- 1$ to the power of $3$ .

$1 - 2 \cdot - 1 + 2 \cdot - 1 - 1$

Step 9.2.1.9.1.1.1.3.4

Multiply $- 2$ by $- 1$ .

$1 + 2 + 2 \cdot - 1 - 1$

Step 9.2.1.9.1.1.1.3.5

Add $1$ and $2$ .

$3 + 2 \cdot - 1 - 1$

Step 9.2.1.9.1.1.1.3.6

Multiply $2$ by $- 1$ .

$3 - 2 - 1$

Step 9.2.1.9.1.1.1.3.7

Subtract $2$ from $3$ .

$1 - 1$

Step 9.2.1.9.1.1.1.3.8

Subtract $1$ from $1$ .

$0$

Step 9.2.1.9.1.1.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 2 x^{3} + 2 x - 1}{x + 1}$

Step 9.2.1.9.1.1.1.5

Divide $x^{4} - 2 x^{3} + 2 x - 1$ by $x + 1$ .

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Step 9.2.1.9.1.1.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

Step 9.2.1.9.1.1.1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

Step 9.2.1.9.1.1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

Step 9.2.1.9.1.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + x^{3}$

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

Step 9.2.1.9.1.1.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

Step 9.2.1.9.1.1.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

Step 9.2.1.9.1.1.1.5.7

Divide the highest order term in the dividend $- 3 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

Step 9.2.1.9.1.1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 9.2.1.9.1.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{3} - 3 x^{2}$

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 9.2.1.9.1.1.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 9.2.1.9.1.1.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

Step 9.2.1.9.1.1.1.5.12

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

Step 9.2.1.9.1.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

Step 9.2.1.9.1.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} + 3 x$

$x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

Step 9.2.1.9.1.1.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

$x$

Step 9.2.1.9.1.1.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

$x$

$1$

Step 9.2.1.9.1.1.1.5.17

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

$x$

$1$

Step 9.2.1.9.1.1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

$x$

$1$

$x$

$1$

Step 9.2.1.9.1.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

$x$

$1$

$x$

$1$

Step 9.2.1.9.1.1.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x$

$1$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$2 x$

$3 x^{2}$

$3 x$

$x$

$1$

$x$

$1$

$0$

Step 9.2.1.9.1.1.1.5.21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} - 3 x^{2} + 3 x - 1$

Step 9.2.1.9.1.1.1.6

Write $x^{4} - 2 x^{3} + 2 x - 1$ as a set of factors.

$(x + 1) ((x - 7) ((x + 1) (x^{3} - 3 x^{2} + 3 x - 1))) = 0$

Step 9.2.1.9.1.1.2

Factor $x^{3} - 3 x^{2} + 3 x - 1$ using the rational roots test.

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Step 9.2.1.9.1.1.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1$

Step 9.2.1.9.1.1.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1$

Step 9.2.1.9.1.1.2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 9.2.1.9.1.1.2.3.1

Substitute $1$ into the polynomial.

$1^{3} - 3 \cdot 1^{2} + 3 \cdot 1 - 1$

Step 9.2.1.9.1.1.2.3.2

Raise $1$ to the power of $3$ .

$1 - 3 \cdot 1^{2} + 3 \cdot 1 - 1$

Step 9.2.1.9.1.1.2.3.3

Raise $1$ to the power of $2$ .

$1 - 3 \cdot 1 + 3 \cdot 1 - 1$

Step 9.2.1.9.1.1.2.3.4

Multiply $- 3$ by $1$ .

$1 - 3 + 3 \cdot 1 - 1$

Step 9.2.1.9.1.1.2.3.5

Subtract $3$ from $1$ .

$- 2 + 3 \cdot 1 - 1$

Step 9.2.1.9.1.1.2.3.6

Multiply $3$ by $1$ .

$- 2 + 3 - 1$

Step 9.2.1.9.1.1.2.3.7

Add $- 2$ and $3$ .

$1 - 1$

Step 9.2.1.9.1.1.2.3.8

Subtract $1$ from $1$ .

$0$

Step 9.2.1.9.1.1.2.4

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 3 x^{2} + 3 x - 1}{x - 1}$

Step 9.2.1.9.1.1.2.5

Divide $x^{3} - 3 x^{2} + 3 x - 1$ by $x - 1$ .

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Step 9.2.1.9.1.1.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

Step 9.2.1.9.1.1.2.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$

Step 9.2.1.9.1.1.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			+	$x^{3}$	-	$x^{2}$

Step 9.2.1.9.1.1.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - x^{2}$

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$

Step 9.2.1.9.1.1.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$

Step 9.2.1.9.1.1.2.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$

Step 9.2.1.9.1.1.2.5.7

Divide the highest order term in the dividend $- 2 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$2 x$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$

Step 9.2.1.9.1.1.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$2 x$

Step 9.2.1.9.1.1.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 2 x$

				$x^{2}$	-	$2 x$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$

Step 9.2.1.9.1.1.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$2 x$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$
							+	$x$

Step 9.2.1.9.1.1.2.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$2 x$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$
							+	$x$	-	$1$

Step 9.2.1.9.1.1.2.5.12

Divide the highest order term in the dividend $x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$2 x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$
							+	$x$	-	$1$

Step 9.2.1.9.1.1.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$
							+	$x$	-	$1$
							+	$x$	-	$1$

Step 9.2.1.9.1.1.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

				$x^{2}$	-	$2 x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$
							+	$x$	-	$1$
							-	$x$	+	$1$

Step 9.2.1.9.1.1.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$2 x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$
							+	$x$	-	$1$
							-	$x$	+	$1$
										$0$

Step 9.2.1.9.1.1.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 2 x + 1$

Step 9.2.1.9.1.1.2.6

Write $x^{3} - 3 x^{2} + 3 x - 1$ as a set of factors.

$(x + 1) ((x - 7) ((x + 1) ((x - 1) (x^{2} - 2 x + 1)))) = 0$

Step 9.2.1.9.1.1.3

Factor using the perfect square rule.

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Step 9.2.1.9.1.1.3.1

Rewrite $1$ as $1^{2}$ .

$(x + 1) ((x - 7) ((x + 1) ((x - 1) (x^{2} - 2 x + 1^{2})))) = 0$

Step 9.2.1.9.1.1.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 9.2.1.9.1.1.3.3

Rewrite the polynomial.

$(x + 1) ((x - 7) ((x + 1) ((x - 1) (x^{2} - 2 \cdot x \cdot 1 + 1^{2})))) = 0$

Step 9.2.1.9.1.1.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$(x + 1) ((x - 7) ((x + 1) ((x - 1) {(x - 1)}^{2}))) = 0$

Step 9.2.1.9.1.1.4

Combine like factors.

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Step 9.2.1.9.1.1.4.1

Raise $x - 1$ to the power of $1$ .

$(x + 1) ((x - 7) ((x + 1) ((x - 1) {(x - 1)}^{2}))) = 0$

Step 9.2.1.9.1.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 1) ((x - 7) ((x + 1) {(x - 1)}^{1 + 2})) = 0$

Step 9.2.1.9.1.1.4.3

Add $1$ and $2$ .

$(x + 1) ((x - 7) ((x + 1) {(x - 1)}^{3})) = 0$

Step 9.2.1.9.1.2

Remove unnecessary parentheses.

$(x + 1) ((x - 7) (x + 1) {(x - 1)}^{3}) = 0$

Step 9.2.1.9.2

Remove unnecessary parentheses.

$(x + 1) (x - 7) (x + 1) {(x - 1)}^{3} = 0$

Step 9.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x - 7 = 0$

$x + 1 = 0$

${(x - 1)}^{3} = 0$

Step 9.2.3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 9.2.3.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 9.2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9.2.4

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 9.2.4.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 9.2.4.2

Add $7$ to both sides of the equation.

$x = 7$

Step 9.2.5

Set ${(x - 1)}^{3}$ equal to $0$ and solve for $x$ .

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Step 9.2.5.1

Set ${(x - 1)}^{3}$ equal to $0$ .

${(x - 1)}^{3} = 0$

Step 9.2.5.2

Solve ${(x - 1)}^{3} = 0$ for $x$ .

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Step 9.2.5.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 9.2.5.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9.2.6

The final solution is all the values that make $(x + 1) (x - 7) (x + 1) {(x - 1)}^{3} = 0$ true.

$x = - 1, 7, 1$

Step 10

The final solution is all the values that make $(x + 1) (x^{6} - 8 x^{5} + 5 x^{4} + 16 x^{3} - 13 x^{2} - 8 x + 7) = 0$ true.

$x = - 1, 7, 1$

Step 11

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 1$

$1 < x < 7$

$x > 7$

Step 12

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 12.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

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Step 12.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 12.1.2

Replace $x$ with $- 4$ in the original inequality.

$(- 4) {(1 - {(- 4)}^{2})}^{3} > 7 {(1 - {(- 4)}^{2})}^{3}$

Step 12.1.3

The left side $13500$ is greater than the right side $- 23625$ , which means that the given statement is always true.

True

Step 12.2

Test a value on the interval $- 1 < x < 1$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $- 1 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 12.2.2

Replace $x$ with $0$ in the original inequality.

$(0) {(1 - {(0)}^{2})}^{3} > 7 {(1 - {(0)}^{2})}^{3}$

Step 12.2.3

The left side $0$ is not greater than the right side $7$ , which means that the given statement is false.

False

Step 12.3

Test a value on the interval $1 < x < 7$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $1 < x < 7$ and see if this value makes the original inequality true.

$x = 4$

Step 12.3.2

Replace $x$ with $4$ in the original inequality.

$(4) {(1 - {(4)}^{2})}^{3} > 7 {(1 - {(4)}^{2})}^{3}$

Step 12.3.3

The left side $- 13500$ is greater than the right side $- 23625$ , which means that the given statement is always true.

True

Step 12.4

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 12.4.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 12.4.2

Replace $x$ with $10$ in the original inequality.

$(10) {(1 - {(10)}^{2})}^{3} > 7 {(1 - {(10)}^{2})}^{3}$

Step 12.4.3

The left side $- 9702990$ is not greater than the right side $- 6792093$ , which means that the given statement is false.

False

Step 12.5

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1$ True

$- 1 < x < 1$ False

$1 < x < 7$ True

$x > 7$ False

$x < - 1$ True

$- 1 < x < 1$ False

$1 < x < 7$ True

$x > 7$ False

Step 13

The solution consists of all of the true intervals.

$x < - 1$ or $1 < x < 7$

Step 14

Convert the inequality to interval notation.

$(- \infty, - 1) \cup (1, 7)$

Step 15