Precalculus Examples

$2 x^{4} - 13 x^{3} + 6 x^{2} + 64 x - 32$

Step 1

Regroup terms.

$64 x - 32 + 2 x^{4} - 13 x^{3} + 6 x^{2}$

Step 2

Factor $32$ out of $64 x - 32$ .

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Step 2.1

Factor $32$ out of $64 x$ .

$32 (2 x) - 32 + 2 x^{4} - 13 x^{3} + 6 x^{2}$

Step 2.2

Factor $32$ out of $- 32$ .

$32 (2 x) + 32 (- 1) + 2 x^{4} - 13 x^{3} + 6 x^{2}$

Step 2.3

Factor $32$ out of $32 (2 x) + 32 (- 1)$ .

$32 (2 x - 1) + 2 x^{4} - 13 x^{3} + 6 x^{2}$

Step 3

Factor $x^{2}$ out of $2 x^{4} - 13 x^{3} + 6 x^{2}$ .

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Step 3.1

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

$32 (2 x - 1) + x^{2} (2 x^{2}) - 13 x^{3} + 6 x^{2}$

Step 3.2

Factor $x^{2}$ out of $- 13 x^{3}$ .

$32 (2 x - 1) + x^{2} (2 x^{2}) + x^{2} (- 13 x) + 6 x^{2}$

Step 3.3

Factor $x^{2}$ out of $6 x^{2}$ .

$32 (2 x - 1) + x^{2} (2 x^{2}) + x^{2} (- 13 x) + x^{2} \cdot 6$

Step 3.4

Factor $x^{2}$ out of $x^{2} (2 x^{2}) + x^{2} (- 13 x)$ .

$32 (2 x - 1) + x^{2} (2 x^{2} - 13 x) + x^{2} \cdot 6$

Step 3.5

Factor $x^{2}$ out of $x^{2} (2 x^{2} - 13 x) + x^{2} \cdot 6$ .

$32 (2 x - 1) + x^{2} (2 x^{2} - 13 x + 6)$

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 $a \cdot c = 2 \cdot 6 = 12$ and whose sum is $b = - 13$ .

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Step 4.1.1.1

Factor $- 13$ out of $- 13 x$ .

$32 (2 x - 1) + x^{2} (2 x^{2} - 13 (x) + 6)$

Step 4.1.1.2

Rewrite $- 13$ as $- 1$ plus $- 12$

$32 (2 x - 1) + x^{2} (2 x^{2} + (- 1 - 12) x + 6)$

Step 4.1.1.3

Apply the distributive property.

$32 (2 x - 1) + x^{2} (2 x^{2} - 1 x - 12 x + 6)$

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.

$32 (2 x - 1) + x^{2} ((2 x^{2} - 1 x) - 12 x + 6)$

Step 4.1.2.2

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

$32 (2 x - 1) + x^{2} (x (2 x - 1) - 6 (2 x - 1))$

Step 4.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$32 (2 x - 1) + x^{2} ((2 x - 1) (x - 6))$

Step 4.2

Remove unnecessary parentheses.

$32 (2 x - 1) + x^{2} (2 x - 1) (x - 6)$

Step 5

Factor $2 x - 1$ out of $32 (2 x - 1) + x^{2} (2 x - 1) (x - 6)$ .

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Step 5.1

Factor $2 x - 1$ out of $32 (2 x - 1)$ .

$(2 x - 1) \cdot 32 + x^{2} (2 x - 1) (x - 6)$

Step 5.2

Factor $2 x - 1$ out of $x^{2} (2 x - 1) (x - 6)$ .

$(2 x - 1) \cdot 32 + (2 x - 1) (x^{2} (x - 6))$

Step 5.3

Factor $2 x - 1$ out of $(2 x - 1) \cdot 32 + (2 x - 1) (x^{2} (x - 6))$ .

$(2 x - 1) (32 + x^{2} (x - 6))$

Step 6

Apply the distributive property.

$(2 x - 1) (32 + x^{2} x + x^{2} \cdot - 6)$

Step 7

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

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Step 7.1

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

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Step 7.1.1

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

$(2 x - 1) (32 + x^{2} x^{1} + x^{2} \cdot - 6)$

Step 7.1.2

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

$(2 x - 1) (32 + x^{2 + 1} + x^{2} \cdot - 6)$

Step 7.2

Add $2$ and $1$ .

$(2 x - 1) (32 + x^{3} + x^{2} \cdot - 6)$

Step 8

Move $- 6$ to the left of $x^{2}$ .

$(2 x - 1) (32 + x^{3} - 6 x^{2})$

Step 9

Reorder terms.

$(2 x - 1) (x^{3} - 6 x^{2} + 32)$

Step 10

Factor.

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Step 10.1

Rewrite $x^{3} - 6 x^{2} + 32$ in a factored form.

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Step 10.1.1

Factor $x^{3} - 6 x^{2} + 32$ using the rational roots test.

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Step 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.

$p = \pm 1, \pm 32, \pm 2, \pm 16, \pm 4, \pm 8$

$q = \pm 1$

Step 10.1.1.2

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

$\pm 1, \pm 32, \pm 2, \pm 16, \pm 4, \pm 8$

Step 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 10.1.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 6 {(- 2)}^{2} + 32$

Step 10.1.1.3.2

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

$- 8 - 6 {(- 2)}^{2} + 32$

Step 10.1.1.3.3

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

$- 8 - 6 \cdot 4 + 32$

Step 10.1.1.3.4

Multiply $- 6$ by $4$ .

$- 8 - 24 + 32$

Step 10.1.1.3.5

Subtract $24$ from $- 8$ .

$- 32 + 32$

Step 10.1.1.3.6

Add $- 32$ and $32$ .

$0$

Step 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.

$\frac{x^{3} - 6 x^{2} + 32}{x + 2}$

Step 10.1.1.5

Divide $x^{3} - 6 x^{2} + 32$ by $x + 2$ .

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Step 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 $0$ .

$x$

$2$

$x^{3}$

$6 x^{2}$

$0 x$

$32$

Step 10.1.1.5.2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$

Step 10.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			+	$x^{3}$	+	$2 x^{2}$

Step 10.1.1.5.4

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$

Step 10.1.1.5.5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$

Step 10.1.1.5.6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$

Step 10.1.1.5.7

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$

Step 10.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					-	$8 x^{2}$	-	$16 x$

Step 10.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 16 x$

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					+	$8 x^{2}$	+	$16 x$

Step 10.1.1.5.10

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

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					+	$8 x^{2}$	+	$16 x$
							+	$16 x$

Step 10.1.1.5.11

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

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					+	$8 x^{2}$	+	$16 x$
							+	$16 x$	+	$32$

Step 10.1.1.5.12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					+	$8 x^{2}$	+	$16 x$
							+	$16 x$	+	$32$

Step 10.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					+	$8 x^{2}$	+	$16 x$
							+	$16 x$	+	$32$
							+	$16 x$	+	$32$

Step 10.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x + 32$

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					+	$8 x^{2}$	+	$16 x$
							+	$16 x$	+	$32$
							-	$16 x$	-	$32$

Step 10.1.1.5.15

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

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	+	$0 x$	+	$32$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	+	$0 x$
					+	$8 x^{2}$	+	$16 x$
							+	$16 x$	+	$32$
							-	$16 x$	-	$32$
										$0$

Step 10.1.1.5.16

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

$x^{2} - 8 x + 16$

Step 10.1.1.6

Write $x^{3} - 6 x^{2} + 32$ as a set of factors.

$(2 x - 1) ((x + 2) (x^{2} - 8 x + 16))$

Step 10.1.2

Factor using the perfect square rule.

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Step 10.1.2.1

Rewrite $16$ as $4^{2}$ .

$(2 x - 1) ((x + 2) (x^{2} - 8 x + 4^{2}))$

Step 10.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 10.1.2.3

Rewrite the polynomial.

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

Step 10.1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 10.2

Remove unnecessary parentheses.

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