Algebra Examples

$x^{4} - 5 x^{3} - 9 x^{2} + 81 x - 108$

Step 1

Regroup terms.

$x^{4} - 9 x^{2} - 5 x^{3} + 81 x - 108$

Step 2

Factor $x^{2}$ out of $x^{4} - 9 x^{2}$ .

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Step 2.1

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

$x^{2} x^{2} - 9 x^{2} - 5 x^{3} + 81 x - 108$

Step 2.2

Factor $x^{2}$ out of $- 9 x^{2}$ .

$x^{2} x^{2} + x^{2} \cdot - 9 - 5 x^{3} + 81 x - 108$

Step 2.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 9$ .

$x^{2} (x^{2} - 9) - 5 x^{3} + 81 x - 108$

Step 3

Rewrite $9$ as $3^{2}$ .

$x^{2} (x^{2} - 3^{2}) - 5 x^{3} + 81 x - 108$

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 $b = 3$ .

$x^{2} ((x + 3) (x - 3)) - 5 x^{3} + 81 x - 108$

Step 4.2

Remove unnecessary parentheses.

$x^{2} (x + 3) (x - 3) - 5 x^{3} + 81 x - 108$

Step 5

Factor $- 5 x^{3} + 81 x - 108$ using the rational roots test.

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Step 5.1

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.

$p = \pm 1, \pm 108, \pm 2, \pm 54, \pm 3, \pm 36, \pm 4, \pm 27, \pm 6, \pm 18, \pm 9, \pm 12$

$q = \pm 1, \pm 5$

Step 5.2

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

$\pm 1, \pm 0.2, \pm 108, \pm 21.6, \pm 2, \pm 0.4, \pm 54, \pm 10.8, \pm 3, \pm 0.6, \pm 36, \pm 7.2, \pm 4, \pm 0.8, \pm 27, \pm 5.4, \pm 6, \pm 1.2, \pm 18, \pm 3.6, \pm 9, \pm 1.8, \pm 12, \pm 2.4$

Step 5.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 5.3.1

Substitute $3$ into the polynomial.

$- 5 \cdot 3^{3} + 81 \cdot 3 - 108$

Step 5.3.2

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

$- 5 \cdot 27 + 81 \cdot 3 - 108$

Step 5.3.3

Multiply $- 5$ by $27$ .

$- 135 + 81 \cdot 3 - 108$

Step 5.3.4

Multiply $81$ by $3$ .

$- 135 + 243 - 108$

Step 5.3.5

Add $- 135$ and $243$ .

$108 - 108$

Step 5.3.6

Subtract $108$ from $108$ .

$0$

Step 5.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 5 x^{3} + 81 x - 108}{x - 3}$

Step 5.5

Divide $- 5 x^{3} + 81 x - 108$ by $x - 3$ .

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Step 5.5.1

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

$x$

$3$

$5 x^{3}$

$0 x^{2}$

$81 x$

$108$

Step 5.5.2

Divide the highest order term in the dividend $- 5 x^{3}$ by the highest order term in divisor $x$ .

				-	$5 x^{2}$
	$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$

Step 5.5.3

Multiply the new quotient term by the divisor.

			-	$5 x^{2}$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			-	$5 x^{3}$	+	$15 x^{2}$

Step 5.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{3} + 15 x^{2}$

			-	$5 x^{2}$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$

Step 5.5.5

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

			-	$5 x^{2}$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$

Step 5.5.6

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

			-	$5 x^{2}$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$

Step 5.5.7

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

			-	$5 x^{2}$	-	$15 x$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$

Step 5.5.8

Multiply the new quotient term by the divisor.

			-	$5 x^{2}$	-	$15 x$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					-	$15 x^{2}$	+	$45 x$

Step 5.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{2} + 45 x$

			-	$5 x^{2}$	-	$15 x$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					+	$15 x^{2}$	-	$45 x$

Step 5.5.10

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

			-	$5 x^{2}$	-	$15 x$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					+	$15 x^{2}$	-	$45 x$
							+	$36 x$

Step 5.5.11

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

			-	$5 x^{2}$	-	$15 x$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					+	$15 x^{2}$	-	$45 x$
							+	$36 x$	-	$108$

Step 5.5.12

Divide the highest order term in the dividend $36 x$ by the highest order term in divisor $x$ .

			-	$5 x^{2}$	-	$15 x$	+	$36$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					+	$15 x^{2}$	-	$45 x$
							+	$36 x$	-	$108$

Step 5.5.13

Multiply the new quotient term by the divisor.

			-	$5 x^{2}$	-	$15 x$	+	$36$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					+	$15 x^{2}$	-	$45 x$
							+	$36 x$	-	$108$
							+	$36 x$	-	$108$

Step 5.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $36 x - 108$

			-	$5 x^{2}$	-	$15 x$	+	$36$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					+	$15 x^{2}$	-	$45 x$
							+	$36 x$	-	$108$
							-	$36 x$	+	$108$

Step 5.5.15

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

			-	$5 x^{2}$	-	$15 x$	+	$36$
$x$	-	$3$	-	$5 x^{3}$	+	$0 x^{2}$	+	$81 x$	-	$108$
			+	$5 x^{3}$	-	$15 x^{2}$
					-	$15 x^{2}$	+	$81 x$
					+	$15 x^{2}$	-	$45 x$
							+	$36 x$	-	$108$
							-	$36 x$	+	$108$
										$0$

Step 5.5.16

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

$- 5 x^{2} - 15 x + 36$

Step 5.6

Write $- 5 x^{3} + 81 x - 108$ as a set of factors.

$x^{2} (x + 3) (x - 3) + (x - 3) (- 5 x^{2} - 15 x + 36)$

Step 6

Factor $x - 3$ out of $x^{2} (x + 3) (x - 3) + (x - 3) (- 5 x^{2} - 15 x + 36)$ .

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Step 6.1

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

$(x - 3) (x^{2} (x + 3)) + (x - 3) (- 5 x^{2} - 15 x + 36)$

Step 6.2

Factor $x - 3$ out of $(x - 3) (x^{2} (x + 3)) + (x - 3) (- 5 x^{2} - 15 x + 36)$ .

$(x - 3) (x^{2} (x + 3) - 5 x^{2} - 15 x + 36)$

Step 7

Apply the distributive property.

$(x - 3) (x^{2} x + x^{2} \cdot 3 - 5 x^{2} - 15 x + 36)$

Step 8

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

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Step 8.1

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

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Step 8.1.1

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

$(x - 3) (x^{2} x^{1} + x^{2} \cdot 3 - 5 x^{2} - 15 x + 36)$

Step 8.1.2

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

$(x - 3) (x^{2 + 1} + x^{2} \cdot 3 - 5 x^{2} - 15 x + 36)$

Step 8.2

Add $2$ and $1$ .

$(x - 3) (x^{3} + x^{2} \cdot 3 - 5 x^{2} - 15 x + 36)$

Step 9

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

$(x - 3) (x^{3} + 3 x^{2} - 5 x^{2} - 15 x + 36)$

Step 10

Subtract $5 x^{2}$ from $3 x^{2}$ .

$(x - 3) (x^{3} - 2 x^{2} - 15 x + 36)$

Step 11

Factor.

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Step 11.1

Rewrite $x^{3} - 2 x^{2} - 15 x + 36$ in a factored form.

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Step 11.1.1

Factor $x^{3} - 2 x^{2} - 15 x + 36$ 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.

$p = \pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6$

$q = \pm 1$

Step 11.1.1.2

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

$\pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6$

Step 11.1.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 11.1.1.3.1

Substitute $3$ into the polynomial.

$3^{3} - 2 \cdot 3^{2} - 15 \cdot 3 + 36$

Step 11.1.1.3.2

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

$27 - 2 \cdot 3^{2} - 15 \cdot 3 + 36$

Step 11.1.1.3.3

Raise $3$ to the power of $2$ .

$27 - 2 \cdot 9 - 15 \cdot 3 + 36$

Step 11.1.1.3.4

Multiply $- 2$ by $9$ .

$27 - 18 - 15 \cdot 3 + 36$

Step 11.1.1.3.5

Subtract $18$ from $27$ .

$9 - 15 \cdot 3 + 36$

Step 11.1.1.3.6

Multiply $- 15$ by $3$ .

$9 - 45 + 36$

Step 11.1.1.3.7

Subtract $45$ from $9$ .

$- 36 + 36$

Step 11.1.1.3.8

Add $- 36$ and $36$ .

$0$

Step 11.1.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 2 x^{2} - 15 x + 36}{x - 3}$

Step 11.1.1.5

Divide $x^{3} - 2 x^{2} - 15 x + 36$ by $x - 3$ .

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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 $0$ .

$x$

$3$

$x^{3}$

$2 x^{2}$

$15 x$

$36$

Step 11.1.1.5.2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$

Step 11.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			+	$x^{3}$	-	$3 x^{2}$

Step 11.1.1.5.4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$

Step 11.1.1.5.5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$

Step 11.1.1.5.6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$

Step 11.1.1.5.7

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$

Step 11.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					+	$x^{2}$	-	$3 x$

Step 11.1.1.5.9

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					-	$x^{2}$	+	$3 x$

Step 11.1.1.5.10

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					-	$x^{2}$	+	$3 x$
							-	$12 x$

Step 11.1.1.5.11

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					-	$x^{2}$	+	$3 x$
							-	$12 x$	+	$36$

Step 11.1.1.5.12

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$x$	-	$12$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					-	$x^{2}$	+	$3 x$
							-	$12 x$	+	$36$

Step 11.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	-	$12$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					-	$x^{2}$	+	$3 x$
							-	$12 x$	+	$36$
							-	$12 x$	+	$36$

Step 11.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x + 36$

				$x^{2}$	+	$x$	-	$12$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					-	$x^{2}$	+	$3 x$
							-	$12 x$	+	$36$
							+	$12 x$	-	$36$

Step 11.1.1.5.15

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

				$x^{2}$	+	$x$	-	$12$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$15 x$	+	$36$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$15 x$
					-	$x^{2}$	+	$3 x$
							-	$12 x$	+	$36$
							+	$12 x$	-	$36$
										$0$

Step 11.1.1.5.16

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

$x^{2} + x - 12$

Step 11.1.1.6

Write $x^{3} - 2 x^{2} - 15 x + 36$ as a set of factors.

$(x - 3) ((x - 3) (x^{2} + x - 12))$

Step 11.1.2

Factor $x^{2} + x - 12$ using the AC method.

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Step 11.1.2.1

Factor $x^{2} + x - 12$ using the AC method.

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Step 11.1.2.1.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 11.1.2.1.2

Write the factored form using these integers.

$(x - 3) ((x - 3) ((x - 3) (x + 4)))$

Step 11.1.2.2

Remove unnecessary parentheses.

$(x - 3) ((x - 3) (x - 3) (x + 4))$

Step 11.1.3

Combine like factors.

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Step 11.1.3.1

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

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

Step 11.1.3.2

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

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

Step 11.1.3.3

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

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

Step 11.1.3.4

Add $1$ and $1$ .

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

Step 11.2

Remove unnecessary parentheses.

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

Step 12

Multiply $x - 3$ by ${(x - 3)}^{2}$ by adding the exponents.

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Step 12.1

Multiply $x - 3$ by ${(x - 3)}^{2}$ .

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.2

Add $1$ and $2$ .

${(x - 3)}^{3} (x + 4)$