Precalculus Examples

x^{4} - 2 x^{3} - 3 x^{2} + 8 x - 4

$x^{4} - 2 x^{3} - 3 x^{2} + 8 x - 4$

Step 1

Regroup terms.

x^{4} - 2 x^{3} - 3 x^{2} + 8 x - 4

$x^{4} - 2 x^{3} - 3 x^{2} + 8 x - 4$

Step 2

Factor

x^{3}

$x^{3}$ out of

x^{4} - 2 x^{3}

$x^{4} - 2 x^{3}$ .

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Step 2.1

Factor

x^{3}

$x^{3}$ out of

x^{4}

$x^{4}$ .

x^{3} x - 2 x^{3} - 3 x^{2} + 8 x - 4

$x^{3} x - 2 x^{3} - 3 x^{2} + 8 x - 4$

Step 2.2

Factor

x^{3}

$x^{3}$ out of

- 2 x^{3}

$- 2 x^{3}$ .

x^{3} x + x^{3} \cdot - 2 - 3 x^{2} + 8 x - 4

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

Step 2.3

Factor

x^{3}

$x^{3}$ out of

x^{3} x + x^{3} \cdot - 2

$x^{3} x + x^{3} \cdot - 2$ .

x^{3} (x - 2) - 3 x^{2} + 8 x - 4

$x^{3} (x - 2) - 3 x^{2} + 8 x - 4$

x^{3} (x - 2) - 3 x^{2} + 8 x - 4

$x^{3} (x - 2) - 3 x^{2} + 8 x - 4$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = - 3 \cdot - 4 = 12

$a \cdot c = - 3 \cdot - 4 = 12$ and whose sum is

b = 8

$b = 8$ .

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Step 3.1.1

Factor

8

$8$ out of

8 x

$8 x$ .

x^{3} (x - 2) - 3 x^{2} + 8 (x) - 4

$x^{3} (x - 2) - 3 x^{2} + 8 (x) - 4$

Step 3.1.2

Rewrite

8

$8$ as

2

$2$ plus

6

$6$

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

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

Step 3.1.3

Apply the distributive property.

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

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

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

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

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

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

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

Step 3.2.2

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

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

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

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor,

- 3 x + 2

$- 3 x + 2$ .

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

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

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

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

Step 4

Factor

$x - 2$ out of

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

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Step 4.1

Factor

$x - 2$ out of

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

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

Step 4.2

Factor

$x - 2$ out of

$(- 3 x + 2) (x - 2)$ .

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

Step 4.3

Factor

$x - 2$ out of

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

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

Step 5

Factor.

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Step 5.1

Rewrite

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

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Step 5.1.1

Factor

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

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Step 5.1.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 5.1.1.2

Find every combination of

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

$\pm 1, \pm 2$

Step 5.1.1.3

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 5.1.1.3.1

Substitute

$1$ into the polynomial.

$1^{3} - 3 \cdot 1 + 2$

Step 5.1.1.3.2

Raise

$1$ to the power of

$3$ .

$1 - 3 \cdot 1 + 2$

Step 5.1.1.3.3

Multiply

$- 3$ by

$1$ .

$1 - 3 + 2$

Step 5.1.1.3.4

Subtract

$3$ from

$1$ .

$- 2 + 2$

Step 5.1.1.3.5

Add

$- 2$ and

$2$ .

$0$

Step 5.1.1.4

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.

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

Step 5.1.1.5

Divide

$x^{3} - 3 x + 2$ by

$x - 1$ .

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Step 5.1.1.5.1

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

$0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$3 x$

$2$

Step 5.1.1.5.2

Divide the highest order term in the dividend

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

$x$ .

					$x^{2}$
	$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$

Step 5.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			+	$x^{3}$	-	$x^{2}$

Step 5.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$

Step 5.1.1.5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$

Step 5.1.1.5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$

Step 5.1.1.5.7

Divide the highest order term in the dividend

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

$x$ .

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$

Step 5.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					+	$x^{2}$	-	$x$

Step 5.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

$x^{2} - x$

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$x$

Step 5.1.1.5.10

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

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$x$
							-	$2 x$

Step 5.1.1.5.11

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

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$x$
							-	$2 x$	+	$2$

Step 5.1.1.5.12

Divide the highest order term in the dividend

$- 2 x$ by the highest order term in divisor

$x$ .

				$x^{2}$	+	$x$	-	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$x$
							-	$2 x$	+	$2$

Step 5.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	-	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$x$
							-	$2 x$	+	$2$
							-	$2 x$	+	$2$

Step 5.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

$- 2 x + 2$

				$x^{2}$	+	$x$	-	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$x$
							-	$2 x$	+	$2$
							+	$2 x$	-	$2$

Step 5.1.1.5.15

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

				$x^{2}$	+	$x$	-	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$x$
							-	$2 x$	+	$2$
							+	$2 x$	-	$2$
										$0$

Step 5.1.1.5.16

Since the remander is

$0$ , the final answer is the quotient.

$x^{2} + x - 2$

Step 5.1.1.6

Write

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

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

Step 5.1.2

Factor

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

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Step 5.1.2.1

Factor

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

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Step 5.1.2.1.1

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

$- 2$ and whose sum is

$1$ .

$- 1, 2$

Step 5.1.2.1.2

Write the factored form using these integers.

$(x - 2) ((x - 1) ((x - 1) (x + 2)))$

Step 5.1.2.2

Remove unnecessary parentheses.

$(x - 2) ((x - 1) (x - 1) (x + 2))$

Step 5.1.3

Combine like factors.

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Step 5.1.3.1

Raise

$x - 1$ to the power of

$1$ .

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

Step 5.1.3.2

Raise

$x - 1$ to the power of

$1$ .

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

Step 5.1.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.1.3.4

Add

$1$ and

$1$ .

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

Step 5.2

Remove unnecessary parentheses.

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