Algebra Examples

$f (x) = - 2 x^{4} + 13 x^{3} - 21 x^{2} + 2 x + 8$

Step 1

Regroup terms.

$f (x) = - 2 x^{4} + 2 x + 13 x^{3} - 21 x^{2} + 8$

Step 2

Factor $- 2 x$ out of $- 2 x^{4} + 2 x$ .

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Step 2.1

Factor $- 2 x$ out of $- 2 x^{4}$ .

$f (x) = - 2 x \cdot x^{3} + 2 x + 13 x^{3} - 21 x^{2} + 8$

Step 2.2

Factor $- 2 x$ out of $2 x$ .

$f (x) = - 2 x \cdot x^{3} - 2 x \cdot - 1 + 13 x^{3} - 21 x^{2} + 8$

Step 2.3

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

$f (x) = - 2 x (x^{3} - 1) + 13 x^{3} - 21 x^{2} + 8$

Step 3

Rewrite $1$ as $1^{3}$ .

$f (x) = - 2 x (x^{3} - 1^{3}) + 13 x^{3} - 21 x^{2} + 8$

Step 4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - i^{3} = (a - i) (a^{2} + a i + i^{2})$ where $a = x$ and $i = 1$ .

$f (x) = - 2 x ((x - 1) (x^{2} + x \cdot 1 + 1^{2})) + 13 x^{3} - 21 x^{2} + 8$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $x$ by $1$ .

$f (x) = - 2 x ((x - 1) (x^{2} + x + 1^{2})) + 13 x^{3} - 21 x^{2} + 8$

Step 5.1.2

One to any power is one.

$f (x) = - 2 x ((x - 1) (x^{2} + x + 1)) + 13 x^{3} - 21 x^{2} + 8$

Step 5.2

Remove unnecessary parentheses.

$f (x) = - 2 x (x - 1) (x^{2} + x + 1) + 13 x^{3} - 21 x^{2} + 8$

Step 6

Factor $13 x^{3} - 21 x^{2} + 8$ using the rational roots test.

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Step 6.1

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.

$f (x) = p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1, \pm 13$

Step 6.2

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

$f (x) = \pm 1, \pm 0.\overline{076923}, \pm 8, \pm 0.\overline{615384}, \pm 2, \pm 0.\overline{153846}, \pm 4, \pm 0.\overline{307692}$

Step 6.3

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 6.3.1

Substitute $1$ into the polynomial.

$f (x) = 13 \cdot 1^{3} - 21 \cdot 1^{2} + 8$

Step 6.3.2

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

$f (x) = 13 \cdot 1 - 21 \cdot 1^{2} + 8$

Step 6.3.3

Multiply $13$ by $1$ .

$f (x) = 13 - 21 \cdot 1^{2} + 8$

Step 6.3.4

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

$f (x) = 13 - 21 \cdot 1 + 8$

Step 6.3.5

Multiply $- 21$ by $1$ .

$f (x) = 13 - 21 + 8$

Step 6.3.6

Subtract $21$ from $13$ .

$f (x) = - 8 + 8$

Step 6.3.7

Add $- 8$ and $8$ .

$f (x) = 0$

Step 6.4

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.

$f (x) = \frac{13 x^{3} - 21 x^{2} + 8}{x - 1}$

Step 6.5

Divide $13 x^{3} - 21 x^{2} + 8$ by $x - 1$ .

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Step 6.5.1

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

$f (x) =$

Step 6.5.2

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

$f (x) =$

Step 6.5.3

Multiply the new quotient term by the divisor.

$f (x) =$

Step 6.5.4

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

$f (x) =$

Step 6.5.5

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

$f (x) =$

Step 6.5.6

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

$f (x) =$

Step 6.5.7

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

$f (x) =$

Step 6.5.8

Multiply the new quotient term by the divisor.

$f (x) =$

Step 6.5.9

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

$f (x) =$

Step 6.5.10

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

$f (x) =$

Step 6.5.11

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

$f (x) =$

Step 6.5.12

Divide the highest order term in the dividend $- 8 x$ by the highest order term in divisor $x$ .

$f (x) =$

Step 6.5.13

Multiply the new quotient term by the divisor.

$f (x) =$

Step 6.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x + 8$

$f (x) =$

Step 6.5.15

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

$f (x) =$

Step 6.5.16

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

$f (x) = 13 x^{2} - 8 x - 8$

Step 6.6

Write $13 x^{3} - 21 x^{2} + 8$ as a set of factors.

$f (x) = - 2 x (x - 1) (x^{2} + x + 1) + (x - 1) (13 x^{2} - 8 x - 8)$

Step 7

Factor $x - 1$ out of $- 2 x (x - 1) (x^{2} + x + 1) + (x - 1) (13 x^{2} - 8 x - 8)$ .

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Step 7.1

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

$f (x) = (x - 1) (- 2 x (x^{2} + x + 1)) + (x - 1) (13 x^{2} - 8 x - 8)$

Step 7.2

Factor $x - 1$ out of $(x - 1) (- 2 x (x^{2} + x + 1)) + (x - 1) (13 x^{2} - 8 x - 8)$ .

$f (x) = (x - 1) (- 2 x (x^{2} + x + 1) + 13 x^{2} - 8 x - 8)$

Step 8

Apply the distributive property.

$f (x) = (x - 1) (- 2 x \cdot x^{2} - 2 x \cdot x - 2 x \cdot 1 + 13 x^{2} - 8 x - 8)$

Step 9

Simplify.

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Step 9.1

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

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Step 9.1.1

Move $x^{2}$ .

$f (x) = (x - 1) (- 2 (x^{2} x) - 2 x \cdot x - 2 x \cdot 1 + 13 x^{2} - 8 x - 8)$

Step 9.1.2

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

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Step 9.1.2.1

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

$f (x) = (x - 1) (- 2 (x^{2} x) - 2 x \cdot x - 2 x \cdot 1 + 13 x^{2} - 8 x - 8)$

Step 9.1.2.2

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

$f (x) = (x - 1) (- 2 x^{2 + 1} - 2 x \cdot x - 2 x \cdot 1 + 13 x^{2} - 8 x - 8)$

Step 9.1.3

Add $2$ and $1$ .

$f (x) = (x - 1) (- 2 x^{3} - 2 x \cdot x - 2 x \cdot 1 + 13 x^{2} - 8 x - 8)$

Step 9.2

Multiply $x$ by $x$ by adding the exponents.

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Step 9.2.1

Move $x$ .

$f (x) = (x - 1) (- 2 x^{3} - 2 (x \cdot x) - 2 x \cdot 1 + 13 x^{2} - 8 x - 8)$

Step 9.2.2

Multiply $x$ by $x$ .

$f (x) = (x - 1) (- 2 x^{3} - 2 x^{2} - 2 x \cdot 1 + 13 x^{2} - 8 x - 8)$

Step 9.3

Multiply $- 2$ by $1$ .

$f (x) = (x - 1) (- 2 x^{3} - 2 x^{2} - 2 x + 13 x^{2} - 8 x - 8)$

Step 10

Add $- 2 x^{2}$ and $13 x^{2}$ .

$f (x) = (x - 1) (- 2 x^{3} + 11 x^{2} - 2 x - 8 x - 8)$

Step 11

Subtract $8 x$ from $- 2 x$ .

$f (x) = (x - 1) (- 2 x^{3} + 11 x^{2} - 10 x - 8)$

Step 12

Factor.

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Step 12.1

Rewrite $- 2 x^{3} + 11 x^{2} - 10 x - 8$ in a factored form.

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Step 12.1.1

Factor $- 2 x^{3} + 11 x^{2} - 10 x - 8$ using the rational roots test.

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Step 12.1.1.1

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.

$f (x) = p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1, \pm 2$

Step 12.1.1.2

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

$f (x) = \pm 1, \pm 0.5, \pm 8, \pm 4, \pm 2$

Step 12.1.1.3

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

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Step 12.1.1.3.1

Substitute $- 0.5$ into the polynomial.

$f (x) = - 2 {(- 0.5)}^{3} + 11 {(- 0.5)}^{2} - 10 \cdot - 0.5 - 8$

Step 12.1.1.3.2

Raise $- 0.5$ to the power of $3$ .

$f (x) = - 2 \cdot - 0.125 + 11 {(- 0.5)}^{2} - 10 \cdot - 0.5 - 8$

Step 12.1.1.3.3

Multiply $- 2$ by $- 0.125$ .

$f (x) = 0.25 + 11 {(- 0.5)}^{2} - 10 \cdot - 0.5 - 8$

Step 12.1.1.3.4

Raise $- 0.5$ to the power of $2$ .

$f (x) = 0.25 + 11 \cdot 0.25 - 10 \cdot - 0.5 - 8$

Step 12.1.1.3.5

Multiply $11$ by $0.25$ .

$f (x) = 0.25 + 2.75 - 10 \cdot - 0.5 - 8$

Step 12.1.1.3.6

Add $0.25$ and $2.75$ .

$f (x) = 3 - 10 \cdot - 0.5 - 8$

Step 12.1.1.3.7

Multiply $- 10$ by $- 0.5$ .

$f (x) = 3 + 5 - 8$

Step 12.1.1.3.8

Add $3$ and $5$ .

$f (x) = 8 - 8$

Step 12.1.1.3.9

Subtract $8$ from $8$ .

$f (x) = 0$

Step 12.1.1.4

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$f (x) = \frac{- 2 x^{3} + 11 x^{2} - 10 x - 8}{2 x + 1}$

Step 12.1.1.5

Divide $- 2 x^{3} + 11 x^{2} - 10 x - 8$ by $2 x + 1$ .

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Step 12.1.1.5.1

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

$f (x) =$

Step 12.1.1.5.2

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

$f (x) =$

Step 12.1.1.5.3

Multiply the new quotient term by the divisor.

$f (x) =$

Step 12.1.1.5.4

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

$f (x) =$

Step 12.1.1.5.5

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

$f (x) =$

Step 12.1.1.5.6

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

$f (x) =$

Step 12.1.1.5.7

Divide the highest order term in the dividend $12 x^{2}$ by the highest order term in divisor $2 x$ .

$f (x) =$

Step 12.1.1.5.8

Multiply the new quotient term by the divisor.

$f (x) =$

Step 12.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{2} + 6 x$

$f (x) =$

Step 12.1.1.5.10

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

$f (x) =$

Step 12.1.1.5.11

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

$f (x) =$

Step 12.1.1.5.12

Divide the highest order term in the dividend $- 16 x$ by the highest order term in divisor $2 x$ .

$f (x) =$

Step 12.1.1.5.13

Multiply the new quotient term by the divisor.

$f (x) =$

Step 12.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x - 8$

$f (x) =$

Step 12.1.1.5.15

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

$f (x) =$

Step 12.1.1.5.16

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

$f (x) = - x^{2} + 6 x - 8$

Step 12.1.1.6

Write $- 2 x^{3} + 11 x^{2} - 10 x - 8$ as a set of factors.

$f (x) = (x - 1) ((2 x + 1) (- x^{2} + 6 x - 8))$

Step 12.1.2

Factor by grouping.

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Step 12.1.2.1

Factor by grouping.

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Step 12.1.2.1.1

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 8 = 8$ and whose sum is $i = 6$ .

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Step 12.1.2.1.1.1

Factor $6$ out of $6 x$ .

$f (x) = (x - 1) ((2 x + 1) (- x^{2} + 6 (x) - 8))$

Step 12.1.2.1.1.2

Rewrite $6$ as $2$ plus $4$

$f (x) = (x - 1) ((2 x + 1) (- x^{2} + (2 + 4) x - 8))$

Step 12.1.2.1.1.3

Apply the distributive property.

$f (x) = (x - 1) ((2 x + 1) (- x^{2} + 2 x + 4 x - 8))$

Step 12.1.2.1.2

Factor out the greatest common factor from each group.

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Step 12.1.2.1.2.1

Group the first two terms and the last two terms.

$f (x) = (x - 1) ((2 x + 1) ((- x^{2} + 2 x) + 4 x - 8))$

Step 12.1.2.1.2.2

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

$f (x) = (x - 1) ((2 x + 1) (x (- x + 2) - 4 (- x + 2)))$

Step 12.1.2.1.3

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

$f (x) = (x - 1) ((2 x + 1) ((- x + 2) (x - 4)))$

Step 12.1.2.2

Remove unnecessary parentheses.

$f (x) = (x - 1) ((2 x + 1) (- x + 2) (x - 4))$

Step 12.2

Remove unnecessary parentheses.

$f (x) = (x - 1) (2 x + 1) (- x + 2) (x - 4)$