Factor -8x+16+8x^3-31x^2

$- 8 x + 16 + 8 x^{3} - 31 x^{2}$

Step 1

Reorder terms.

$8 x^{3} - 31 x^{2} - 8 x + 16$

Step 2

Factor $8 x^{3} - 31 x^{2} - 8 x + 16$ using the rational roots test.

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.

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

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

$\pm 1, \pm 0.125, \pm 0.5, \pm 0.25, \pm 16, \pm 2, \pm 8, \pm 4$

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

Substitute $4$ into the polynomial.

$8 \cdot 4^{3} - 31 \cdot 4^{2} - 8 \cdot 4 + 16$

Raise $4$ to the power of $3$ .

$8 \cdot 64 - 31 \cdot 4^{2} - 8 \cdot 4 + 16$

Multiply $8$ by $64$ .

$512 - 31 \cdot 4^{2} - 8 \cdot 4 + 16$

Raise $4$ to the power of $2$ .

$512 - 31 \cdot 16 - 8 \cdot 4 + 16$

Multiply $- 31$ by $16$ .

$512 - 496 - 8 \cdot 4 + 16$

Subtract $496$ from $512$ .

$16 - 8 \cdot 4 + 16$

Multiply $- 8$ by $4$ .

$16 - 32 + 16$

Subtract $32$ from $16$ .

$- 16 + 16$

Add $- 16$ and $16$ .

$0$

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{8 x^{3} - 31 x^{2} - 8 x + 16}{x - 4}$

Divide $8 x^{3} - 31 x^{2} - 8 x + 16$ by $x - 4$ .

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

$x$

$4$

$8 x^{3}$

$31 x^{2}$

$8 x$

$16$

Divide the highest order term in the dividend $8 x^{3}$ by the highest order term in divisor $x$ .

					$8 x^{2}$
	$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$

Multiply the new quotient term by the divisor.

				$8 x^{2}$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			+	$8 x^{3}$	-	$32 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} - 32 x^{2}$

				$8 x^{2}$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$

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

				$8 x^{2}$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$

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

				$8 x^{2}$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$

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

				$8 x^{2}$	+	$x$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$

Multiply the new quotient term by the divisor.

				$8 x^{2}$	+	$x$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					+	$x^{2}$	-	$4 x$

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

				$8 x^{2}$	+	$x$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					-	$x^{2}$	+	$4 x$

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

				$8 x^{2}$	+	$x$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					-	$x^{2}$	+	$4 x$
							-	$4 x$

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

				$8 x^{2}$	+	$x$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					-	$x^{2}$	+	$4 x$
							-	$4 x$	+	$16$

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

				$8 x^{2}$	+	$x$	-	$4$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					-	$x^{2}$	+	$4 x$
							-	$4 x$	+	$16$

Multiply the new quotient term by the divisor.

				$8 x^{2}$	+	$x$	-	$4$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					-	$x^{2}$	+	$4 x$
							-	$4 x$	+	$16$
							-	$4 x$	+	$16$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x + 16$

				$8 x^{2}$	+	$x$	-	$4$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					-	$x^{2}$	+	$4 x$
							-	$4 x$	+	$16$
							+	$4 x$	-	$16$

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

				$8 x^{2}$	+	$x$	-	$4$
$x$	-	$4$		$8 x^{3}$	-	$31 x^{2}$	-	$8 x$	+	$16$
			-	$8 x^{3}$	+	$32 x^{2}$
					+	$x^{2}$	-	$8 x$
					-	$x^{2}$	+	$4 x$
							-	$4 x$	+	$16$
							+	$4 x$	-	$16$
										$0$

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

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

Write $8 x^{3} - 31 x^{2} - 8 x + 16$ as a set of factors.

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

Factor -8x+16+8x^3-31x^2

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