Precalculus Examples

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

Step 1

Simplify $x^{2} (x + 1)$ .

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Step 1.1

Rewrite.

$0 + 0 + x^{2} (x + 1) = 4 (2 x + 3)$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Apply the distributive property.

$x^{2} x + x^{2} \cdot 1 = 4 (2 x + 3)$

Step 1.4

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

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Step 1.4.1

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

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Step 1.4.1.1

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

$x^{2} x^{1} + x^{2} \cdot 1 = 4 (2 x + 3)$

Step 1.4.1.2

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

$x^{2 + 1} + x^{2} \cdot 1 = 4 (2 x + 3)$

Step 1.4.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot 1 = 4 (2 x + 3)$

Step 1.5

Multiply $x^{2}$ by $1$ .

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

Step 2

Simplify $4 (2 x + 3)$ .

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply.

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Step 2.2.1

Multiply $2$ by $4$ .

$x^{3} + x^{2} = 8 x + 4 \cdot 3$

Step 2.2.2

Multiply $4$ by $3$ .

$x^{3} + x^{2} = 8 x + 12$

Step 3

Subtract $8 x$ from both sides of the equation.

$x^{3} + x^{2} - 8 x = 12$

Step 4

Subtract $12$ from both sides of the equation.

$x^{3} + x^{2} - 8 x - 12 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $x^{3} + x^{2} - 8 x - 12$ using the rational roots test.

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Step 5.1.1

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.

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

$q = \pm 1$

Step 5.1.2

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 5.1.3

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 5.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} + {(- 2)}^{2} - 8 \cdot - 2 - 12$

Step 5.1.3.2

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

$- 8 + {(- 2)}^{2} - 8 \cdot - 2 - 12$

Step 5.1.3.3

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

$- 8 + 4 - 8 \cdot - 2 - 12$

Step 5.1.3.4

Add $- 8$ and $4$ .

$- 4 - 8 \cdot - 2 - 12$

Step 5.1.3.5

Multiply $- 8$ by $- 2$ .

$- 4 + 16 - 12$

Step 5.1.3.6

Add $- 4$ and $16$ .

$12 - 12$

Step 5.1.3.7

Subtract $12$ from $12$ .

$0$

Step 5.1.4

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.

$\frac{x^{3} + x^{2} - 8 x - 12}{x + 2}$

Step 5.1.5

Divide $x^{3} + x^{2} - 8 x - 12$ by $x + 2$ .

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Step 5.1.5.1

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

$x$

$2$

$x^{3}$

$x^{2}$

$8 x$

$12$

Step 5.1.5.2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$

Step 5.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			+	$x^{3}$	+	$2 x^{2}$

Step 5.1.5.4

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$

Step 5.1.5.5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$

Step 5.1.5.6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$

Step 5.1.5.7

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

				$x^{2}$	-	$x$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$

Step 5.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					-	$x^{2}$	-	$2 x$

Step 5.1.5.9

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

				$x^{2}$	-	$x$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$2 x$

Step 5.1.5.10

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

				$x^{2}$	-	$x$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$2 x$
							-	$6 x$

Step 5.1.5.11

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

				$x^{2}$	-	$x$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$2 x$
							-	$6 x$	-	$12$

Step 5.1.5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$x$	-	$6$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$2 x$
							-	$6 x$	-	$12$

Step 5.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$6$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$2 x$
							-	$6 x$	-	$12$
							-	$6 x$	-	$12$

Step 5.1.5.14

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

				$x^{2}$	-	$x$	-	$6$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$2 x$
							-	$6 x$	-	$12$
							+	$6 x$	+	$12$

Step 5.1.5.15

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

				$x^{2}$	-	$x$	-	$6$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$2 x$
							-	$6 x$	-	$12$
							+	$6 x$	+	$12$
										$0$

Step 5.1.5.16

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

$x^{2} - x - 6$

Step 5.1.6

Write $x^{3} + x^{2} - 8 x - 12$ as a set of factors.

$(x + 2) (x^{2} - x - 6) = 0$

Step 5.2

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

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Step 5.2.1

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

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Step 5.2.1.1

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

$- 3, 2$

Step 5.2.1.2

Write the factored form using these integers.

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

Step 5.2.2

Remove unnecessary parentheses.

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

Step 6

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

$x + 2 = 0$

$x - 3 = 0$

$x + 2 = 0$

Step 7

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

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Step 7.1

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

$x + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8

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

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Step 8.1

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

$x - 3 = 0$

Step 8.2

Add $3$ to both sides of the equation.

$x = 3$

Step 9

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

$x = - 2, 3$

Step 10