Algebra Examples

${(\frac{18}{x} + x)}^{2} + 2 (\frac{18}{x} + x) = 99$

Step 1

Simplify ${(\frac{18}{x} + x)}^{2} + 2 (\frac{18}{x} + x)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

${(\frac{18}{x} + x)}^{2} + 2 \frac{18}{x} + 2 x = 99$

Step 1.1.2

Multiply $2 \frac{18}{x}$ .

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Step 1.1.2.1

Combine $2$ and $\frac{18}{x}$ .

${(\frac{18}{x} + x)}^{2} + \frac{2 \cdot 18}{x} + 2 x = 99$

Step 1.1.2.2

Multiply $2$ by $18$ .

${(\frac{18}{x} + x)}^{2} + \frac{36}{x} + 2 x = 99$

Step 1.2

To write ${(\frac{18}{x} + x)}^{2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{{(\frac{18}{x} + x)}^{2} x}{x} + \frac{36}{x} + 2 x = 99$

Step 1.3

Combine the numerators over the common denominator.

$\frac{{(\frac{18}{x} + x)}^{2} x + 36}{x} + 2 x = 99$

Step 1.4

Simplify each term.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Simplify each term.

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Step 1.4.1.1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{{(\frac{18}{x} + \frac{x \cdot x}{x})}^{2} x + 36}{x} + 2 x = 99$

Step 1.4.1.1.2

Combine the numerators over the common denominator.

$\frac{{(\frac{18 + x \cdot x}{x})}^{2} x + 36}{x} + 2 x = 99$

Step 1.4.1.1.3

Multiply $x$ by $x$ .

$\frac{{(\frac{18 + x^{2}}{x})}^{2} x + 36}{x} + 2 x = 99$

Step 1.4.1.1.4

Apply the product rule to $\frac{18 + x^{2}}{x}$ .

$\frac{\frac{{(18 + x^{2})}^{2}}{x^{2}} x + 36}{x} + 2 x = 99$

Step 1.4.1.1.5

Cancel the common factor of $x$ .

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Step 1.4.1.1.5.1

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

$\frac{\frac{{(18 + x^{2})}^{2}}{x \cdot x} x + 36}{x} + 2 x = 99$

Step 1.4.1.1.5.2

Cancel the common factor.

$\frac{\frac{{(18 + x^{2})}^{2}}{x \cdot x} x + 36}{x} + 2 x = 99$

Step 1.4.1.1.5.3

Rewrite the expression.

$\frac{\frac{{(18 + x^{2})}^{2}}{x} + 36}{x} + 2 x = 99$

Step 1.4.1.2

To write $36$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{\frac{{(18 + x^{2})}^{2}}{x} + \frac{36 x}{x}}{x} + 2 x = 99$

Step 1.4.1.3

Combine the numerators over the common denominator.

$\frac{\frac{{(18 + x^{2})}^{2} + 36 x}{x}}{x} + 2 x = 99$

Step 1.4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{{(18 + x^{2})}^{2} + 36 x}{x} \cdot \frac{1}{x} + 2 x = 99$

Step 1.4.3

Multiply $\frac{{(18 + x^{2})}^{2} + 36 x}{x} \cdot \frac{1}{x}$ .

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Step 1.4.3.1

Multiply $\frac{{(18 + x^{2})}^{2} + 36 x}{x}$ by $\frac{1}{x}$ .

$\frac{{(18 + x^{2})}^{2} + 36 x}{x \cdot x} + 2 x = 99$

Step 1.4.3.2

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

$\frac{{(18 + x^{2})}^{2} + 36 x}{x^{1} x} + 2 x = 99$

Step 1.4.3.3

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

$\frac{{(18 + x^{2})}^{2} + 36 x}{x^{1} x^{1}} + 2 x = 99$

Step 1.4.3.4

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

$\frac{{(18 + x^{2})}^{2} + 36 x}{x^{1 + 1}} + 2 x = 99$

Step 1.4.3.5

Add $1$ and $1$ .

$\frac{{(18 + x^{2})}^{2} + 36 x}{x^{2}} + 2 x = 99$

Step 2

Factor each term.

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Step 2.1

Rewrite ${(18 + x^{2})}^{2}$ as $(18 + x^{2}) (18 + x^{2})$ .

$\frac{(18 + x^{2}) (18 + x^{2}) + 36 x}{x^{2}} + 2 x = 99$

Step 2.2

Expand $(18 + x^{2}) (18 + x^{2})$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\frac{18 (18 + x^{2}) + x^{2} (18 + x^{2}) + 36 x}{x^{2}} + 2 x = 99$

Step 2.2.2

Apply the distributive property.

$\frac{18 \cdot 18 + 18 x^{2} + x^{2} (18 + x^{2}) + 36 x}{x^{2}} + 2 x = 99$

Step 2.2.3

Apply the distributive property.

$\frac{18 \cdot 18 + 18 x^{2} + x^{2} \cdot 18 + x^{2} x^{2} + 36 x}{x^{2}} + 2 x = 99$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $18$ by $18$ .

$\frac{324 + 18 x^{2} + x^{2} \cdot 18 + x^{2} x^{2} + 36 x}{x^{2}} + 2 x = 99$

Step 2.3.1.2

Move $18$ to the left of $x^{2}$ .

$\frac{324 + 18 x^{2} + 18 x^{2} + x^{2} x^{2} + 36 x}{x^{2}} + 2 x = 99$

Step 2.3.1.3

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

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Step 2.3.1.3.1

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

$\frac{324 + 18 x^{2} + 18 x^{2} + x^{2 + 2} + 36 x}{x^{2}} + 2 x = 99$

Step 2.3.1.3.2

Add $2$ and $2$ .

$\frac{324 + 18 x^{2} + 18 x^{2} + x^{4} + 36 x}{x^{2}} + 2 x = 99$

Step 2.3.2

Add $18 x^{2}$ and $18 x^{2}$ .

$\frac{324 + 36 x^{2} + x^{4} + 36 x}{x^{2}} + 2 x = 99$

Step 2.4

Reorder terms.

$\frac{x^{4} + 36 x^{2} + 36 x + 324}{x^{2}} + 2 x = 99$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1, 1$

Step 3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 4

Multiply each term in $\frac{x^{4} + 36 x^{2} + 36 x + 324}{x^{2}} + 2 x = 99$ by $x^{2}$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{x^{4} + 36 x^{2} + 36 x + 324}{x^{2}} + 2 x = 99$ by $x^{2}$ .

$\frac{x^{4} + 36 x^{2} + 36 x + 324}{x^{2}} x^{2} + 2 x \cdot x^{2} = 99 x^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $x^{2}$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{x^{4} + 36 x^{2} + 36 x + 324}{x^{2}} x^{2} + 2 x \cdot x^{2} = 99 x^{2}$

Step 4.2.1.1.2

Rewrite the expression.

$x^{4} + 36 x^{2} + 36 x + 324 + 2 x \cdot x^{2} = 99 x^{2}$

Step 4.2.1.2

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

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Step 4.2.1.2.1

Move $x^{2}$ .

$x^{4} + 36 x^{2} + 36 x + 324 + 2 (x^{2} x) = 99 x^{2}$

Step 4.2.1.2.2

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

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Step 4.2.1.2.2.1

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

$x^{4} + 36 x^{2} + 36 x + 324 + 2 (x^{2} x^{1}) = 99 x^{2}$

Step 4.2.1.2.2.2

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

$x^{4} + 36 x^{2} + 36 x + 324 + 2 x^{2 + 1} = 99 x^{2}$

Step 4.2.1.2.3

Add $2$ and $1$ .

$x^{4} + 36 x^{2} + 36 x + 324 + 2 x^{3} = 99 x^{2}$

Step 5

Solve the equation.

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Step 5.1

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

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Step 5.1.1

Subtract $99 x^{2}$ from both sides of the equation.

$x^{4} + 36 x^{2} + 36 x + 324 + 2 x^{3} - 99 x^{2} = 0$

Step 5.1.2

Subtract $99 x^{2}$ from $36 x^{2}$ .

$x^{4} - 63 x^{2} + 36 x + 324 + 2 x^{3} = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Regroup terms.

$36 x + 324 + x^{4} - 63 x^{2} + 2 x^{3} = 0$

Step 5.2.2

Factor $36$ out of $36 x + 324$ .

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Step 5.2.2.1

Factor $36$ out of $36 x$ .

$36 (x) + 324 + x^{4} - 63 x^{2} + 2 x^{3} = 0$

Step 5.2.2.2

Factor $36$ out of $324$ .

$36 x + 36 \cdot 9 + x^{4} - 63 x^{2} + 2 x^{3} = 0$

Step 5.2.2.3

Factor $36$ out of $36 x + 36 \cdot 9$ .

$36 (x + 9) + x^{4} - 63 x^{2} + 2 x^{3} = 0$

Step 5.2.3

Factor $x^{2}$ out of $x^{4} - 63 x^{2} + 2 x^{3}$ .

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Step 5.2.3.1

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

$36 (x + 9) + x^{2} x^{2} - 63 x^{2} + 2 x^{3} = 0$

Step 5.2.3.2

Factor $x^{2}$ out of $- 63 x^{2}$ .

$36 (x + 9) + x^{2} x^{2} + x^{2} \cdot - 63 + 2 x^{3} = 0$

Step 5.2.3.3

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

$36 (x + 9) + x^{2} x^{2} + x^{2} \cdot - 63 + x^{2} (2 x) = 0$

Step 5.2.3.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 63$ .

$36 (x + 9) + x^{2} (x^{2} - 63) + x^{2} (2 x) = 0$

Step 5.2.3.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 63) + x^{2} (2 x)$ .

$36 (x + 9) + x^{2} (x^{2} - 63 + 2 x) = 0$

Step 5.2.4

Factor.

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Step 5.2.4.1

Factor $x^{2} - 63 + 2 x$ using the AC method.

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Step 5.2.4.1.1

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 $- 63$ and whose sum is $2$ .

$- 7, 9$

Step 5.2.4.1.2

Write the factored form using these integers.

$36 (x + 9) + x^{2} ((x - 7) (x + 9)) = 0$

Step 5.2.4.2

Remove unnecessary parentheses.

$36 (x + 9) + x^{2} (x - 7) (x + 9) = 0$

Step 5.2.5

Factor $x + 9$ out of $36 (x + 9) + x^{2} (x - 7) (x + 9)$ .

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Step 5.2.5.1

Factor $x + 9$ out of $36 (x + 9)$ .

$(x + 9) \cdot 36 + x^{2} (x - 7) (x + 9) = 0$

Step 5.2.5.2

Factor $x + 9$ out of $x^{2} (x - 7) (x + 9)$ .

$(x + 9) \cdot 36 + (x + 9) (x^{2} (x - 7)) = 0$

Step 5.2.5.3

Factor $x + 9$ out of $(x + 9) \cdot 36 + (x + 9) (x^{2} (x - 7))$ .

$(x + 9) (36 + x^{2} (x - 7)) = 0$

Step 5.2.6

Apply the distributive property.

$(x + 9) (36 + x^{2} x + x^{2} \cdot - 7) = 0$

Step 5.2.7

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

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Step 5.2.7.1

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

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Step 5.2.7.1.1

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

$(x + 9) (36 + x^{2} x + x^{2} \cdot - 7) = 0$

Step 5.2.7.1.2

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

$(x + 9) (36 + x^{2 + 1} + x^{2} \cdot - 7) = 0$

Step 5.2.7.2

Add $2$ and $1$ .

$(x + 9) (36 + x^{3} + x^{2} \cdot - 7) = 0$

Step 5.2.8

Move $- 7$ to the left of $x^{2}$ .

$(x + 9) (36 + x^{3} - 7 x^{2}) = 0$

Step 5.2.9

Reorder terms.

$(x + 9) (x^{3} - 7 x^{2} + 36) = 0$

Step 5.2.10

Factor.

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Step 5.2.10.1

Rewrite $x^{3} - 7 x^{2} + 36$ in a factored form.

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Step 5.2.10.1.1

Factor $x^{3} - 7 x^{2} + 36$ using the rational roots test.

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Step 5.2.10.1.1.1

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 5.2.10.1.1.2

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 5.2.10.1.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 5.2.10.1.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 7 {(- 2)}^{2} + 36$

Step 5.2.10.1.1.3.2

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

$- 8 - 7 {(- 2)}^{2} + 36$

Step 5.2.10.1.1.3.3

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

$- 8 - 7 \cdot 4 + 36$

Step 5.2.10.1.1.3.4

Multiply $- 7$ by $4$ .

$- 8 - 28 + 36$

Step 5.2.10.1.1.3.5

Subtract $28$ from $- 8$ .

$- 36 + 36$

Step 5.2.10.1.1.3.6

Add $- 36$ and $36$ .

$0$

Step 5.2.10.1.1.4

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

$\frac{x^{3} - 7 x^{2} + 36}{x + 2}$

Step 5.2.10.1.1.5

Divide $x^{3} - 7 x^{2} + 36$ by $x + 2$ .

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Step 5.2.10.1.1.5.1

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

$x$

$2$

$x^{3}$

$7 x^{2}$

$0 x$

$36$

Step 5.2.10.1.1.5.2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$

Step 5.2.10.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			+	$x^{3}$	+	$2 x^{2}$

Step 5.2.10.1.1.5.4

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$

Step 5.2.10.1.1.5.5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$

Step 5.2.10.1.1.5.6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$

Step 5.2.10.1.1.5.7

Divide the highest order term in the dividend $- 9 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$9 x$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$

Step 5.2.10.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	-	$18 x$

Step 5.2.10.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{2} - 18 x$

				$x^{2}$	-	$9 x$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	+	$18 x$

Step 5.2.10.1.1.5.10

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

				$x^{2}$	-	$9 x$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	+	$18 x$
							+	$18 x$

Step 5.2.10.1.1.5.11

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

				$x^{2}$	-	$9 x$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	+	$18 x$
							+	$18 x$	+	$36$

Step 5.2.10.1.1.5.12

Divide the highest order term in the dividend $18 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$9 x$	+	$18$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	+	$18 x$
							+	$18 x$	+	$36$

Step 5.2.10.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$	+	$18$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	+	$18 x$
							+	$18 x$	+	$36$
							+	$18 x$	+	$36$

Step 5.2.10.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $18 x + 36$

				$x^{2}$	-	$9 x$	+	$18$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	+	$18 x$
							+	$18 x$	+	$36$
							-	$18 x$	-	$36$

Step 5.2.10.1.1.5.15

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

				$x^{2}$	-	$9 x$	+	$18$
$x$	+	$2$		$x^{3}$	-	$7 x^{2}$	+	$0 x$	+	$36$
			-	$x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	+	$18 x$
							+	$18 x$	+	$36$
							-	$18 x$	-	$36$
										$0$

Step 5.2.10.1.1.5.16

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

$x^{2} - 9 x + 18$

Step 5.2.10.1.1.6

Write $x^{3} - 7 x^{2} + 36$ as a set of factors.

$(x + 9) ((x + 2) (x^{2} - 9 x + 18)) = 0$

Step 5.2.10.1.2

Factor $x^{2} - 9 x + 18$ using the AC method.

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Step 5.2.10.1.2.1

Factor $x^{2} - 9 x + 18$ using the AC method.

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Step 5.2.10.1.2.1.1

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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Step 5.2.10.1.2.1.2

Write the factored form using these integers.

$(x + 9) ((x + 2) ((x - 6) (x - 3))) = 0$

Step 5.2.10.1.2.2

Remove unnecessary parentheses.

$(x + 9) ((x + 2) (x - 6) (x - 3)) = 0$

Step 5.2.10.2

Remove unnecessary parentheses.

$(x + 9) (x + 2) (x - 6) (x - 3) = 0$

Step 5.3

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

$x + 9 = 0$

$x + 2 = 0$

$x - 6 = 0$

$x - 3 = 0$

Step 5.4

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 5.4.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 5.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 5.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.6

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 5.6.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 5.6.2

Add $6$ to both sides of the equation.

$x = 6$

Step 5.7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 5.7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5.8

The final solution is all the values that make $(x + 9) (x + 2) (x - 6) (x - 3) = 0$ true.

$x = - 9, - 2, 6, 3$