Algebra Examples

$\log_{2} (x + 3) + \log_{2} (x^{2} - 3 x - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} ((x + 3) (x^{2} - 3 x - 2)) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.2

Expand $(x + 3) (x^{2} - 3 x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$\log_{2} (x \cdot x^{2} + x (- 3 x) + x \cdot - 2 + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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Step 1.3.1.1.1

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

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Step 1.3.1.1.1.1

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

$\log_{2} (x \cdot x^{2} + x (- 3 x) + x \cdot - 2 + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.1.1.2

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

$\log_{2} (x^{1 + 2} + x (- 3 x) + x \cdot - 2 + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.1.2

Add $1$ and $2$ .

$\log_{2} (x^{3} + x (- 3 x) + x \cdot - 2 + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$\log_{2} (x^{3} - 3 x \cdot x + x \cdot - 2 + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.3

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

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Step 1.3.1.3.1

Move $x$ .

$\log_{2} (x^{3} - 3 (x \cdot x) + x \cdot - 2 + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.3.2

Multiply $x$ by $x$ .

$\log_{2} (x^{3} - 3 x^{2} + x \cdot - 2 + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.4

Move $- 2$ to the left of $x$ .

$\log_{2} (x^{3} - 3 x^{2} - 2 \cdot x + 3 x^{2} + 3 (- 3 x) + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.5

Multiply $- 3$ by $3$ .

$\log_{2} (x^{3} - 3 x^{2} - 2 x + 3 x^{2} - 9 x + 3 \cdot - 2) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.1.6

Multiply $3$ by $- 2$ .

$\log_{2} (x^{3} - 3 x^{2} - 2 x + 3 x^{2} - 9 x - 6) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.2

Simplify by adding terms.

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Step 1.3.2.1

Combine the opposite terms in $x^{3} - 3 x^{2} - 2 x + 3 x^{2} - 9 x - 6$ .

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Step 1.3.2.1.1

Add $- 3 x^{2}$ and $3 x^{2}$ .

$\log_{2} (x^{3} - 2 x + 0 - 9 x - 6) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.2.1.2

Add $x^{3} - 2 x$ and $0$ .

$\log_{2} (x^{3} - 2 x - 9 x - 6) = \log_{2} (x^{2} + x - 6) + 2$

Step 1.3.2.2

Subtract $9 x$ from $- 2 x$ .

$\log_{2} (x^{3} - 11 x - 6) = \log_{2} (x^{2} + x - 6) + 2$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\log_{2} (x^{3} - 11 x - 6) - \log_{2} (x^{2} + x - 6) = 2$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{2} (\frac{x^{3} - 11 x - 6}{x^{2} + x - 6}) = 2$

Step 4

Factor $x^{2} + x - 6$ using the AC method.

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Step 4.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 4.2

Write the factored form using these integers.

$\log_{2} (\frac{x^{3} - 11 x - 6}{(x - 2) (x + 3)}) = 2$

Step 5

Rewrite $\log_{2} (\frac{x^{3} - 11 x - 6}{(x - 2) (x + 3)}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$2^{2} = \frac{x^{3} - 11 x - 6}{(x - 2) (x + 3)}$

Step 6

Cross multiply to remove the fraction.

$x^{3} - 11 x - 6 = 2^{2} ((x - 2) (x + 3))$

Step 7

Simplify $2^{2} ((x - 2) (x + 3))$ .

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Step 7.1

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

$x^{3} - 11 x - 6 = 4 ((x - 2) (x + 3))$

Step 7.2

Expand $(x - 2) (x + 3)$ using the FOIL Method.

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Step 7.2.1

Apply the distributive property.

$x^{3} - 11 x - 6 = 4 (x (x + 3) - 2 (x + 3))$

Step 7.2.2

Apply the distributive property.

$x^{3} - 11 x - 6 = 4 (x \cdot x + x \cdot 3 - 2 (x + 3))$

Step 7.2.3

Apply the distributive property.

$x^{3} - 11 x - 6 = 4 (x \cdot x + x \cdot 3 - 2 x - 2 \cdot 3)$

Step 7.3

Simplify and combine like terms.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Multiply $x$ by $x$ .

$x^{3} - 11 x - 6 = 4 (x^{2} + x \cdot 3 - 2 x - 2 \cdot 3)$

Step 7.3.1.2

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

$x^{3} - 11 x - 6 = 4 (x^{2} + 3 \cdot x - 2 x - 2 \cdot 3)$

Step 7.3.1.3

Multiply $- 2$ by $3$ .

$x^{3} - 11 x - 6 = 4 (x^{2} + 3 x - 2 x - 6)$

Step 7.3.2

Subtract $2 x$ from $3 x$ .

$x^{3} - 11 x - 6 = 4 (x^{2} + x - 6)$

Step 7.4

Apply the distributive property.

$x^{3} - 11 x - 6 = 4 x^{2} + 4 x + 4 \cdot - 6$

Step 7.5

Multiply $4$ by $- 6$ .

$x^{3} - 11 x - 6 = 4 x^{2} + 4 x - 24$

Step 8

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

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Step 8.1

Subtract $4 x^{2}$ from both sides of the equation.

$x^{3} - 11 x - 6 - 4 x^{2} = 4 x - 24$

Step 8.2

Subtract $4 x$ from both sides of the equation.

$x^{3} - 11 x - 6 - 4 x^{2} - 4 x = - 24$

Step 8.3

Subtract $4 x$ from $- 11 x$ .

$x^{3} - 15 x - 6 - 4 x^{2} = - 24$

Step 9

Factor the left side of the equation.

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Step 9.1

Reorder terms.

$x^{3} - 4 x^{2} - 15 x - 6 = - 24$

Step 9.2

Factor $x^{3} - 4 x^{2} - 15 x - 6$ using the rational roots test.

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Step 9.2.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

Step 9.2.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 9.2.3

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 9.2.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 4 {(- 2)}^{2} - 15 \cdot - 2 - 6$

Step 9.2.3.2

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

$- 8 - 4 {(- 2)}^{2} - 15 \cdot - 2 - 6$

Step 9.2.3.3

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

$- 8 - 4 \cdot 4 - 15 \cdot - 2 - 6$

Step 9.2.3.4

Multiply $- 4$ by $4$ .

$- 8 - 16 - 15 \cdot - 2 - 6$

Step 9.2.3.5

Subtract $16$ from $- 8$ .

$- 24 - 15 \cdot - 2 - 6$

Step 9.2.3.6

Multiply $- 15$ by $- 2$ .

$- 24 + 30 - 6$

Step 9.2.3.7

Add $- 24$ and $30$ .

$6 - 6$

Step 9.2.3.8

Subtract $6$ from $6$ .

$0$

Step 9.2.4

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.

$\frac{x^{3} - 4 x^{2} - 15 x - 6}{x + 2}$

Step 9.2.5

Divide $x^{3} - 4 x^{2} - 15 x - 6$ by $x + 2$ .

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Step 9.2.5.1

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

$x$

$2$

$x^{3}$

$4 x^{2}$

$15 x$

$6$

Step 9.2.5.2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$

Step 9.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			+	$x^{3}$	+	$2 x^{2}$

Step 9.2.5.4

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$

Step 9.2.5.5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$

Step 9.2.5.6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$

Step 9.2.5.7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$

Step 9.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					-	$6 x^{2}$	-	$12 x$

Step 9.2.5.9

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					+	$6 x^{2}$	+	$12 x$

Step 9.2.5.10

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					+	$6 x^{2}$	+	$12 x$
							-	$3 x$

Step 9.2.5.11

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					+	$6 x^{2}$	+	$12 x$
							-	$3 x$	-	$6$

Step 9.2.5.12

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

				$x^{2}$	-	$6 x$	-	$3$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					+	$6 x^{2}$	+	$12 x$
							-	$3 x$	-	$6$

Step 9.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	-	$3$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					+	$6 x^{2}$	+	$12 x$
							-	$3 x$	-	$6$
							-	$3 x$	-	$6$

Step 9.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x - 6$

				$x^{2}$	-	$6 x$	-	$3$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					+	$6 x^{2}$	+	$12 x$
							-	$3 x$	-	$6$
							+	$3 x$	+	$6$

Step 9.2.5.15

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

				$x^{2}$	-	$6 x$	-	$3$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	-	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$15 x$
					+	$6 x^{2}$	+	$12 x$
							-	$3 x$	-	$6$
							+	$3 x$	+	$6$
										$0$

Step 9.2.5.16

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

$x^{2} - 6 x - 3$

Step 9.2.6

Write $x^{3} - 4 x^{2} - 15 x - 6$ as a set of factors.

$(x + 2) (x^{2} - 6 x - 3) = - 24$

Step 10

Simplify $(x + 2) (x^{2} - 6 x - 3)$ .

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Step 10.1

Expand $(x + 2) (x^{2} - 6 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{2} + x (- 6 x) + x \cdot - 3 + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2

Simplify terms.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

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

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Step 10.2.1.1.1

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

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Step 10.2.1.1.1.1

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

$x^{1} x^{2} + x (- 6 x) + x \cdot - 3 + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2.1.1.1.2

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

$x^{1 + 2} + x (- 6 x) + x \cdot - 3 + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2.1.1.2

Add $1$ and $2$ .

$x^{3} + x (- 6 x) + x \cdot - 3 + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2.1.2

Rewrite using the commutative property of multiplication.

$x^{3} - 6 x \cdot x + x \cdot - 3 + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2.1.3

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

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Step 10.2.1.3.1

Move $x$ .

$x^{3} - 6 (x \cdot x) + x \cdot - 3 + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2.1.3.2

Multiply $x$ by $x$ .

$x^{3} - 6 x^{2} + x \cdot - 3 + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2.1.4

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

$x^{3} - 6 x^{2} - 3 \cdot x + 2 x^{2} + 2 (- 6 x) + 2 \cdot - 3 = - 24$

Step 10.2.1.5

Multiply $- 6$ by $2$ .

$x^{3} - 6 x^{2} - 3 x + 2 x^{2} - 12 x + 2 \cdot - 3 = - 24$

Step 10.2.1.6

Multiply $2$ by $- 3$ .

$x^{3} - 6 x^{2} - 3 x + 2 x^{2} - 12 x - 6 = - 24$

Step 10.2.2

Simplify by adding terms.

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Step 10.2.2.1

Add $- 6 x^{2}$ and $2 x^{2}$ .

$x^{3} - 4 x^{2} - 3 x - 12 x - 6 = - 24$

Step 10.2.2.2

Subtract $12 x$ from $- 3 x$ .

$x^{3} - 4 x^{2} - 15 x - 6 = - 24$

Step 11

Add $24$ to both sides of the equation.

$x^{3} - 4 x^{2} - 15 x - 6 + 24 = 0$

Step 12

Add $- 6$ and $24$ .

$x^{3} - 4 x^{2} - 15 x + 18 = 0$

Step 13

Factor the left side of the equation.

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Step 13.1

Factor $x^{3} - 4 x^{2} - 15 x + 18$ using the rational roots test.

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Step 13.1.1

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.

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

$q = \pm 1$

Step 13.1.2

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

$\pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

Step 13.1.3

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 13.1.3.1

Substitute $1$ into the polynomial.

$1^{3} - 4 \cdot 1^{2} - 15 \cdot 1 + 18$

Step 13.1.3.2

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

$1 - 4 \cdot 1^{2} - 15 \cdot 1 + 18$

Step 13.1.3.3

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

$1 - 4 \cdot 1 - 15 \cdot 1 + 18$

Step 13.1.3.4

Multiply $- 4$ by $1$ .

$1 - 4 - 15 \cdot 1 + 18$

Step 13.1.3.5

Subtract $4$ from $1$ .

$- 3 - 15 \cdot 1 + 18$

Step 13.1.3.6

Multiply $- 15$ by $1$ .

$- 3 - 15 + 18$

Step 13.1.3.7

Subtract $15$ from $- 3$ .

$- 18 + 18$

Step 13.1.3.8

Add $- 18$ and $18$ .

$0$

Step 13.1.4

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.

$\frac{x^{3} - 4 x^{2} - 15 x + 18}{x - 1}$

Step 13.1.5

Divide $x^{3} - 4 x^{2} - 15 x + 18$ by $x - 1$ .

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Step 13.1.5.1

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

$x$

$1$

$x^{3}$

$4 x^{2}$

$15 x$

$18$

Step 13.1.5.2

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$

Step 13.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			+	$x^{3}$	-	$x^{2}$

Step 13.1.5.4

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$

Step 13.1.5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$

Step 13.1.5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$

Step 13.1.5.7

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

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$

Step 13.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					-	$3 x^{2}$	+	$3 x$

Step 13.1.5.9

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

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					+	$3 x^{2}$	-	$3 x$

Step 13.1.5.10

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

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					+	$3 x^{2}$	-	$3 x$
							-	$18 x$

Step 13.1.5.11

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

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					+	$3 x^{2}$	-	$3 x$
							-	$18 x$	+	$18$

Step 13.1.5.12

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

				$x^{2}$	-	$3 x$	-	$18$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					+	$3 x^{2}$	-	$3 x$
							-	$18 x$	+	$18$

Step 13.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	-	$18$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					+	$3 x^{2}$	-	$3 x$
							-	$18 x$	+	$18$
							-	$18 x$	+	$18$

Step 13.1.5.14

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

				$x^{2}$	-	$3 x$	-	$18$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					+	$3 x^{2}$	-	$3 x$
							-	$18 x$	+	$18$
							+	$18 x$	-	$18$

Step 13.1.5.15

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

				$x^{2}$	-	$3 x$	-	$18$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$15 x$
					+	$3 x^{2}$	-	$3 x$
							-	$18 x$	+	$18$
							+	$18 x$	-	$18$
										$0$

Step 13.1.5.16

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

$x^{2} - 3 x - 18$

Step 13.1.6

Write $x^{3} - 4 x^{2} - 15 x + 18$ as a set of factors.

$(x - 1) (x^{2} - 3 x - 18) = 0$

Step 13.2

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 13.2.1

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 13.2.1.1

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 $- 3$ .

$- 6, 3$

Step 13.2.1.2

Write the factored form using these integers.

$(x - 1) ((x - 6) (x + 3)) = 0$

Step 13.2.2

Remove unnecessary parentheses.

$(x - 1) (x - 6) (x + 3) = 0$

Step 14

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

$x - 1 = 0$

$x - 6 = 0$

$x + 3 = 0$

Step 15

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 15.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 15.2

Add $1$ to both sides of the equation.

$x = 1$

Step 16

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

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Step 16.1

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

$x - 6 = 0$

Step 16.2

Add $6$ to both sides of the equation.

$x = 6$

Step 17

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

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Step 17.1

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

$x + 3 = 0$

Step 17.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 18

The final solution is all the values that make $(x - 1) (x - 6) (x + 3) = 0$ true.

$x = 1, 6, - 3$

Step 19

Exclude the solutions that do not make $\log_{2} (x + 3) + \log_{2} (x^{2} - 3 x - 2) = \log_{2} (x^{2} + x - 6) + 2$ true.

$x = 6$