Algebra Examples

$2 x^{4} + 23 x^{3} + 60 x^{2} - 125 x - 500$

Step 1

Regroup terms.

$- 125 x - 500 + 2 x^{4} + 23 x^{3} + 60 x^{2}$

Step 2

Factor $- 125$ out of $- 125 x - 500$ .

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Step 2.1

Factor $- 125$ out of $- 125 x$ .

$- 125 (x) - 500 + 2 x^{4} + 23 x^{3} + 60 x^{2}$

Step 2.2

Factor $- 125$ out of $- 500$ .

$- 125 (x) - 125 (4) + 2 x^{4} + 23 x^{3} + 60 x^{2}$

Step 2.3

Factor $- 125$ out of $- 125 (x) - 125 (4)$ .

$- 125 (x + 4) + 2 x^{4} + 23 x^{3} + 60 x^{2}$

Step 3

Factor $x^{2}$ out of $2 x^{4} + 23 x^{3} + 60 x^{2}$ .

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Step 3.1

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

$- 125 (x + 4) + x^{2} (2 x^{2}) + 23 x^{3} + 60 x^{2}$

Step 3.2

Factor $x^{2}$ out of $23 x^{3}$ .

$- 125 (x + 4) + x^{2} (2 x^{2}) + x^{2} (23 x) + 60 x^{2}$

Step 3.3

Factor $x^{2}$ out of $60 x^{2}$ .

$- 125 (x + 4) + x^{2} (2 x^{2}) + x^{2} (23 x) + x^{2} \cdot 60$

Step 3.4

Factor $x^{2}$ out of $x^{2} (2 x^{2}) + x^{2} (23 x)$ .

$- 125 (x + 4) + x^{2} (2 x^{2} + 23 x) + x^{2} \cdot 60$

Step 3.5

Factor $x^{2}$ out of $x^{2} (2 x^{2} + 23 x) + x^{2} \cdot 60$ .

$- 125 (x + 4) + x^{2} (2 x^{2} + 23 x + 60)$

Step 4

Factor.

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Step 4.1

Factor by grouping.

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Step 4.1.1

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 $a \cdot c = 2 \cdot 60 = 120$ and whose sum is $b = 23$ .

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Step 4.1.1.1

Factor $23$ out of $23 x$ .

$- 125 (x + 4) + x^{2} (2 x^{2} + 23 (x) + 60)$

Step 4.1.1.2

Rewrite $23$ as $8$ plus $15$

$- 125 (x + 4) + x^{2} (2 x^{2} + (8 + 15) x + 60)$

Step 4.1.1.3

Apply the distributive property.

$- 125 (x + 4) + x^{2} (2 x^{2} + 8 x + 15 x + 60)$

Step 4.1.2

Factor out the greatest common factor from each group.

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Step 4.1.2.1

Group the first two terms and the last two terms.

$- 125 (x + 4) + x^{2} ((2 x^{2} + 8 x) + 15 x + 60)$

Step 4.1.2.2

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

$- 125 (x + 4) + x^{2} (2 x (x + 4) + 15 (x + 4))$

Step 4.1.3

Factor the polynomial by factoring out the greatest common factor, $x + 4$ .

$- 125 (x + 4) + x^{2} ((x + 4) (2 x + 15))$

Step 4.2

Remove unnecessary parentheses.

$- 125 (x + 4) + x^{2} (x + 4) (2 x + 15)$

Step 5

Factor $x + 4$ out of $- 125 (x + 4) + x^{2} (x + 4) (2 x + 15)$ .

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Step 5.1

Factor $x + 4$ out of $- 125 (x + 4)$ .

$(x + 4) \cdot - 125 + x^{2} (x + 4) (2 x + 15)$

Step 5.2

Factor $x + 4$ out of $x^{2} (x + 4) (2 x + 15)$ .

$(x + 4) \cdot - 125 + (x + 4) (x^{2} (2 x + 15))$

Step 5.3

Factor $x + 4$ out of $(x + 4) \cdot - 125 + (x + 4) (x^{2} (2 x + 15))$ .

$(x + 4) (- 125 + x^{2} (2 x + 15))$

Step 6

Apply the distributive property.

$(x + 4) (- 125 + x^{2} (2 x) + x^{2} \cdot 15)$

Step 7

Rewrite using the commutative property of multiplication.

$(x + 4) (- 125 + 2 x^{2} x + x^{2} \cdot 15)$

Step 8

Move $15$ to the left of $x^{2}$ .

$(x + 4) (- 125 + 2 x^{2} x + 15 \cdot x^{2})$

Step 9

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

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Step 9.1

Move $x$ .

$(x + 4) (- 125 + 2 (x \cdot x^{2}) + 15 \cdot x^{2})$

Step 9.2

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

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Step 9.2.1

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

$(x + 4) (- 125 + 2 (x^{1} x^{2}) + 15 \cdot x^{2})$

Step 9.2.2

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

$(x + 4) (- 125 + 2 x^{1 + 2} + 15 \cdot x^{2})$

Step 9.3

Add $1$ and $2$ .

$(x + 4) (- 125 + 2 x^{3} + 15 \cdot x^{2})$

$(x + 4) (- 125 + 2 x^{3} + 15 x^{2})$

Step 10

Reorder terms.

$(x + 4) (2 x^{3} + 15 x^{2} - 125)$

Step 11

Factor.

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Step 11.1

Rewrite $2 x^{3} + 15 x^{2} - 125$ in a factored form.

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Step 11.1.1

Factor $2 x^{3} + 15 x^{2} - 125$ using the rational roots test.

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Step 11.1.1.1

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.

$p = \pm 1, \pm 125, \pm 5, \pm 25$

$q = \pm 1, \pm 2$

Step 11.1.1.2

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

$\pm 1, \pm 0.5, \pm 125, \pm 62.5, \pm 5, \pm 2.5, \pm 25, \pm 12.5$

Step 11.1.1.3

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

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Step 11.1.1.3.1

Substitute $- 5$ into the polynomial.

$2 {(- 5)}^{3} + 15 {(- 5)}^{2} - 125$

Step 11.1.1.3.2

Raise $- 5$ to the power of $3$ .

$2 \cdot - 125 + 15 {(- 5)}^{2} - 125$

Step 11.1.1.3.3

Multiply $2$ by $- 125$ .

$- 250 + 15 {(- 5)}^{2} - 125$

Step 11.1.1.3.4

Raise $- 5$ to the power of $2$ .

$- 250 + 15 \cdot 25 - 125$

Step 11.1.1.3.5

Multiply $15$ by $25$ .

$- 250 + 375 - 125$

Step 11.1.1.3.6

Add $- 250$ and $375$ .

$125 - 125$

Step 11.1.1.3.7

Subtract $125$ from $125$ .

$0$

Step 11.1.1.4

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 15 x^{2} - 125}{x + 5}$

Step 11.1.1.5

Divide $2 x^{3} + 15 x^{2} - 125$ by $x + 5$ .

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Step 11.1.1.5.1

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

$x$

$5$

$2 x^{3}$

$15 x^{2}$

$0 x$

$125$

Step 11.1.1.5.2

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

					$2 x^{2}$
	$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$

Step 11.1.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			+	$2 x^{3}$	+	$10 x^{2}$

Step 11.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 10 x^{2}$

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$

Step 11.1.1.5.5

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

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$

Step 11.1.1.5.6

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

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$

Step 11.1.1.5.7

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$5 x$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$

Step 11.1.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$5 x$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$25 x$

Step 11.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 25 x$

				$2 x^{2}$	+	$5 x$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$25 x$

Step 11.1.1.5.10

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

				$2 x^{2}$	+	$5 x$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$25 x$
							-	$25 x$

Step 11.1.1.5.11

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

				$2 x^{2}$	+	$5 x$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$25 x$
							-	$25 x$	-	$125$

Step 11.1.1.5.12

Divide the highest order term in the dividend $- 25 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$5 x$	-	$25$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$25 x$
							-	$25 x$	-	$125$

Step 11.1.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$5 x$	-	$25$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$25 x$
							-	$25 x$	-	$125$
							-	$25 x$	-	$125$

Step 11.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 25 x - 125$

				$2 x^{2}$	+	$5 x$	-	$25$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$25 x$
							-	$25 x$	-	$125$
							+	$25 x$	+	$125$

Step 11.1.1.5.15

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

				$2 x^{2}$	+	$5 x$	-	$25$
$x$	+	$5$		$2 x^{3}$	+	$15 x^{2}$	+	$0 x$	-	$125$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$25 x$
							-	$25 x$	-	$125$
							+	$25 x$	+	$125$
										$0$

Step 11.1.1.5.16

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

$2 x^{2} + 5 x - 25$

Step 11.1.1.6

Write $2 x^{3} + 15 x^{2} - 125$ as a set of factors.

$(x + 4) ((x + 5) (2 x^{2} + 5 x - 25))$

Step 11.1.2

Factor by grouping.

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Step 11.1.2.1

Factor by grouping.

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Step 11.1.2.1.1

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 $a \cdot c = 2 \cdot - 25 = - 50$ and whose sum is $b = 5$ .

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Step 11.1.2.1.1.1

Factor $5$ out of $5 x$ .

$(x + 4) ((x + 5) (2 x^{2} + 5 (x) - 25))$

Step 11.1.2.1.1.2

Rewrite $5$ as $- 5$ plus $10$

$(x + 4) ((x + 5) (2 x^{2} + (- 5 + 10) x - 25))$

Step 11.1.2.1.1.3

Apply the distributive property.

$(x + 4) ((x + 5) (2 x^{2} - 5 x + 10 x - 25))$

Step 11.1.2.1.2

Factor out the greatest common factor from each group.

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Step 11.1.2.1.2.1

Group the first two terms and the last two terms.

$(x + 4) ((x + 5) ((2 x^{2} - 5 x) + 10 x - 25))$

Step 11.1.2.1.2.2

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

$(x + 4) ((x + 5) (x (2 x - 5) + 5 (2 x - 5)))$

Step 11.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 5$ .

$(x + 4) ((x + 5) ((2 x - 5) (x + 5)))$

Step 11.1.2.2

Remove unnecessary parentheses.

$(x + 4) ((x + 5) (2 x - 5) (x + 5))$

Step 11.2

Remove unnecessary parentheses.

$(x + 4) (x + 5) (2 x - 5) (x + 5)$

Step 12

Combine exponents.

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Step 12.1

Raise $x + 5$ to the power of $1$ .

$(x + 4) ({(x + 5)}^{1} (x + 5)) (2 x - 5)$

Step 12.2

Raise $x + 5$ to the power of $1$ .

$(x + 4) ({(x + 5)}^{1} {(x + 5)}^{1}) (2 x - 5)$

Step 12.3

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

$(x + 4) {(x + 5)}^{1 + 1} (2 x - 5)$

Step 12.4

Add $1$ and $1$ .

$(x + 4) {(x + 5)}^{2} (2 x - 5)$