Algebra Examples

$- x^{5} + 52 x^{3} + 2 x^{2} - 147 x - 98$

Step 1

Regroup terms.

$2 x^{2} - 98 - x^{5} + 52 x^{3} - 147 x$

Step 2

Factor $2$ out of $2 x^{2} - 98$ .

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Step 2.1

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

$2 (x^{2}) - 98 - x^{5} + 52 x^{3} - 147 x$

Step 2.2

Factor $2$ out of $- 98$ .

$2 x^{2} + 2 \cdot - 49 - x^{5} + 52 x^{3} - 147 x$

Step 2.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 49$ .

$2 (x^{2} - 49) - x^{5} + 52 x^{3} - 147 x$

Step 3

Rewrite $49$ as $7^{2}$ .

$2 (x^{2} - 7^{2}) - x^{5} + 52 x^{3} - 147 x$

Step 4

Factor.

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Step 4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$2 ((x + 7) (x - 7)) - x^{5} + 52 x^{3} - 147 x$

Step 4.2

Remove unnecessary parentheses.

$2 (x + 7) (x - 7) - x^{5} + 52 x^{3} - 147 x$

Step 5

Factor $x$ out of $- x^{5} + 52 x^{3} - 147 x$ .

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Step 5.1

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

$2 (x + 7) (x - 7) + x (- x^{4}) + 52 x^{3} - 147 x$

Step 5.2

Factor $x$ out of $52 x^{3}$ .

$2 (x + 7) (x - 7) + x (- x^{4}) + x (52 x^{2}) - 147 x$

Step 5.3

Factor $x$ out of $- 147 x$ .

$2 (x + 7) (x - 7) + x (- x^{4}) + x (52 x^{2}) + x \cdot - 147$

Step 5.4

Factor $x$ out of $x (- x^{4}) + x (52 x^{2})$ .

$2 (x + 7) (x - 7) + x (- x^{4} + 52 x^{2}) + x \cdot - 147$

Step 5.5

Factor $x$ out of $x (- x^{4} + 52 x^{2}) + x \cdot - 147$ .

$2 (x + 7) (x - 7) + x (- x^{4} + 52 x^{2} - 147)$

Step 6

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$2 (x + 7) (x - 7) + x (- {(x^{2})}^{2} + 52 x^{2} - 147)$

Step 7

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$2 (x + 7) (x - 7) + x (- u^{2} + 52 u - 147)$

Step 8

Factor by grouping.

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Step 8.1

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 $a \cdot c = - 1 \cdot - 147 = 147$ and whose sum is $b = 52$ .

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Step 8.1.1

Factor $52$ out of $52 u$ .

$2 (x + 7) (x - 7) + x (- u^{2} + 52 (u) - 147)$

Step 8.1.2

Rewrite $52$ as $3$ plus $49$

$2 (x + 7) (x - 7) + x (- u^{2} + (3 + 49) u - 147)$

Step 8.1.3

Apply the distributive property.

$2 (x + 7) (x - 7) + x (- u^{2} + 3 u + 49 u - 147)$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$2 (x + 7) (x - 7) + x ((- u^{2} + 3 u) + 49 u - 147)$

Step 8.2.2

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

$2 (x + 7) (x - 7) + x (u (- u + 3) - 49 (- u + 3))$

Step 8.3

Factor the polynomial by factoring out the greatest common factor, $- u + 3$ .

$2 (x + 7) (x - 7) + x ((- u + 3) (u - 49))$

Step 9

Replace all occurrences of $u$ with $x^{2}$ .

$2 (x + 7) (x - 7) + x ((- x^{2} + 3) (x^{2} - 49))$

Step 10

Rewrite $49$ as $7^{2}$ .

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

Step 11

Factor.

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Step 11.1

Factor.

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Step 11.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

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

Step 11.1.2

Remove unnecessary parentheses.

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

Step 11.2

Remove unnecessary parentheses.

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 13

Apply the distributive property.

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

Step 14

Rewrite using the commutative property of multiplication.

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

Step 15

Move $3$ to the left of $x$ .

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

Step 16

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

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Step 16.1

Move $x^{2}$ .

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

Step 16.2

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

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Step 16.2.1

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

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

Step 16.2.2

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

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

Step 16.3

Add $2$ and $1$ .

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

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

Step 17

Reorder terms.

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

Step 18

Factor.

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Step 18.1

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

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Step 18.1.1

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

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Step 18.1.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 18.1.1.2

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

$\pm 1, \pm 2$

Step 18.1.1.3

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 18.1.1.3.1

Substitute $- 1$ into the polynomial.

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

Step 18.1.1.3.2

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

$- - 1 + 3 \cdot - 1 + 2$

Step 18.1.1.3.3

Multiply $- 1$ by $- 1$ .

$1 + 3 \cdot - 1 + 2$

Step 18.1.1.3.4

Multiply $3$ by $- 1$ .

$1 - 3 + 2$

Step 18.1.1.3.5

Subtract $3$ from $1$ .

$- 2 + 2$

Step 18.1.1.3.6

Add $- 2$ and $2$ .

$0$

Step 18.1.1.4

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.

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

Step 18.1.1.5

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

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Step 18.1.1.5.1

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

$x$

$1$

$x^{3}$

$0 x^{2}$

$3 x$

$2$

Step 18.1.1.5.2

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

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

Step 18.1.1.5.3

Multiply the new quotient term by the divisor.

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

Step 18.1.1.5.4

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

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

Step 18.1.1.5.5

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

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

Step 18.1.1.5.6

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

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

Step 18.1.1.5.7

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

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

Step 18.1.1.5.8

Multiply the new quotient term by the divisor.

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

Step 18.1.1.5.9

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

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

Step 18.1.1.5.10

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

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

Step 18.1.1.5.11

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

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

Step 18.1.1.5.12

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

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

Step 18.1.1.5.13

Multiply the new quotient term by the divisor.

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

Step 18.1.1.5.14

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

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

Step 18.1.1.5.15

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

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

Step 18.1.1.5.16

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

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

Step 18.1.1.6

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

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

Step 18.1.2

Factor by grouping.

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Step 18.1.2.1

Factor by grouping.

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Step 18.1.2.1.1

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 $a \cdot c = - 1 \cdot 2 = - 2$ and whose sum is $b = 1$ .

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Step 18.1.2.1.1.1

Multiply by $1$ .

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

Step 18.1.2.1.1.2

Rewrite $1$ as $- 1$ plus $2$

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

Step 18.1.2.1.1.3

Apply the distributive property.

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

Step 18.1.2.1.2

Factor out the greatest common factor from each group.

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Step 18.1.2.1.2.1

Group the first two terms and the last two terms.

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

Step 18.1.2.1.2.2

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

$(x + 7) (x - 7) ((x + 1) (x (- x - 1) - 2 (- x - 1)))$

Step 18.1.2.1.3

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

$(x + 7) (x - 7) ((x + 1) ((- x - 1) (x - 2)))$

Step 18.1.2.2

Remove unnecessary parentheses.

$(x + 7) (x - 7) ((x + 1) (- x - 1) (x - 2))$

Step 18.2

Remove unnecessary parentheses.

$(x + 7) (x - 7) (x + 1) (- x - 1) (x - 2)$

Step 19

Combine exponents.

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Step 19.1

Factor $- 1$ out of $x$ .

$(x + 7) (x - 7) (- 1 (- x) + 1) (- x - 1) (x - 2)$

Step 19.2

Rewrite $1$ as $- 1 (- 1)$ .

$(x + 7) (x - 7) (- 1 (- x) - 1 (- 1)) (- x - 1) (x - 2)$

Step 19.3

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

$(x + 7) (x - 7) (- 1 (- x - 1)) (- x - 1) (x - 2)$

Step 19.4

Remove parentheses.

$(x + 7) (x - 7) \cdot - 1 (- x - 1) (- x - 1) (x - 2)$

Step 19.5

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

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

Step 19.6

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

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

Step 19.7

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

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

Step 19.8

Add $1$ and $1$ .

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

Step 20

Factor.

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Step 20.1

Factor out negative.

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

Step 20.2

Remove unnecessary parentheses.

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

Step 21

Factor $- 1$ out of $- x - 1$ .

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Step 21.1

Factor $- 1$ out of $- x$ .

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

Step 21.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 21.3

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

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

Step 22

Factor.

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Step 22.1

Apply the product rule to $- (x + 1)$ .

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

Step 22.2

Remove unnecessary parentheses.

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

Step 23

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 23.1

Move ${(- 1)}^{2}$ .

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

Step 23.2

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

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Step 23.2.1

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

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

Step 23.2.2

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

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

Step 23.3

Add $2$ and $1$ .

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