Precalculus Examples

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

Step 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.

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

$q = \pm 1, \pm 3$

Step 2

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

$\pm 1, \pm 0.\overline{3}, \pm 8, \pm 2.\overline{6}, \pm 2, \pm 0.\overline{6}, \pm 4, \pm 1.\overline{3}$

Step 3

Substitute $- 0.\overline{6}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{6}$ is a root of the polynomial.

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Step 3.1

Substitute $- 0.\overline{6}$ into the polynomial.

$3 {(- 0.\overline{6})}^{3} - 4 {(- 0.\overline{6})}^{2} + 8 \cdot - 0.\overline{6} + 8$

Step 3.2

Raise $- 0.\overline{6}$ to the power of $3$ .

$3 \cdot - 0.\overline{296} - 4 {(- 0.\overline{6})}^{2} + 8 \cdot - 0.\overline{6} + 8$

Step 3.3

Multiply $3$ by $- 0.\overline{296}$ .

$- 0.\overline{8} - 4 {(- 0.\overline{6})}^{2} + 8 \cdot - 0.\overline{6} + 8$

Step 3.4

Raise $- 0.\overline{6}$ to the power of $2$ .

$- 0.\overline{8} - 4 \cdot 0.\overline{4} + 8 \cdot - 0.\overline{6} + 8$

Step 3.5

Multiply $- 4$ by $0.\overline{4}$ .

$- 0.\overline{8} - 1.\overline{7} + 8 \cdot - 0.\overline{6} + 8$

Step 3.6

Subtract $1.\overline{7}$ from $- 0.\overline{8}$ .

$- 2.\overline{6} + 8 \cdot - 0.\overline{6} + 8$

Step 3.7

Multiply $8$ by $- 0.\overline{6}$ .

$- 2.\overline{6} - 5.\overline{3} + 8$

Step 3.8

Subtract $5.\overline{3}$ from $- 2.\overline{6}$ .

$- 8 + 8$

Step 3.9

Add $- 8$ and $8$ .

$0$

Step 4

Since $- 0.\overline{6}$ is a known root, divide the polynomial by $3 x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 5

Divide $3 x^{3} - 4 x^{2} + 8 x + 8$ by $3 x + 2$ .

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Step 5.1

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

$3 x$

$2$

$3 x^{3}$

$4 x^{2}$

$8 x$

$8$

Step 5.2

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

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

Step 5.3

Multiply the new quotient term by the divisor.

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

Step 5.4

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

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

Step 5.5

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

				$x^{2}$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$

Step 5.6

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

				$x^{2}$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$

Step 5.7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $3 x$ .

				$x^{2}$	-	$2 x$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$

Step 5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					-	$6 x^{2}$	-	$4 x$

Step 5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 4 x$

				$x^{2}$	-	$2 x$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					+	$6 x^{2}$	+	$4 x$

Step 5.10

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

				$x^{2}$	-	$2 x$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					+	$6 x^{2}$	+	$4 x$
							+	$12 x$

Step 5.11

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

				$x^{2}$	-	$2 x$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					+	$6 x^{2}$	+	$4 x$
							+	$12 x$	+	$8$

Step 5.12

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	-	$2 x$	+	$4$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					+	$6 x^{2}$	+	$4 x$
							+	$12 x$	+	$8$

Step 5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	+	$4$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					+	$6 x^{2}$	+	$4 x$
							+	$12 x$	+	$8$
							+	$12 x$	+	$8$

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $12 x + 8$

				$x^{2}$	-	$2 x$	+	$4$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					+	$6 x^{2}$	+	$4 x$
							+	$12 x$	+	$8$
							-	$12 x$	-	$8$

Step 5.15

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

				$x^{2}$	-	$2 x$	+	$4$
$3 x$	+	$2$		$3 x^{3}$	-	$4 x^{2}$	+	$8 x$	+	$8$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$8 x$
					+	$6 x^{2}$	+	$4 x$
							+	$12 x$	+	$8$
							-	$12 x$	-	$8$
										$0$

Step 5.16

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

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

Step 6

Write $3 x^{3} - 4 x^{2} + 8 x + 8$ as a set of factors.

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