Factor 32x^4+24x^3-16x^2-8x

$32 x^{4} + 24 x^{3} - 16 x^{2} - 8 x$

Factor $8 x$ out of $32 x^{4} + 24 x^{3} - 16 x^{2} - 8 x$ .

Factor $8 x$ out of $32 x^{4}$ .

$8 x (4 x^{3}) + 24 x^{3} - 16 x^{2} - 8 x$

Factor $8 x$ out of $24 x^{3}$ .

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

Factor $8 x$ out of $- 16 x^{2}$ .

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

Factor $8 x$ out of $- 8 x$ .

$8 x (4 x^{3}) + 8 x (3 x^{2}) + 8 x (- 2 x) + 8 x (- 1)$

Factor $8 x$ out of $8 x (4 x^{3}) + 8 x (3 x^{2})$ .

$8 x (4 x^{3} + 3 x^{2}) + 8 x (- 2 x) + 8 x (- 1)$

Factor $8 x$ out of $8 x (4 x^{3} + 3 x^{2}) + 8 x (- 2 x)$ .

$8 x (4 x^{3} + 3 x^{2} - 2 x) + 8 x (- 1)$

Factor $8 x$ out of $8 x (4 x^{3} + 3 x^{2} - 2 x) + 8 x (- 1)$ .

$8 x (4 x^{3} + 3 x^{2} - 2 x - 1)$

Factor.

Factor $4 x^{3} + 3 x^{2} - 2 x - 1$ 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.

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

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

$\pm 1, \pm 0.25, \pm 0.5$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

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

Raise $- 1$ to the power of $3$ .

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

Multiply $4$ by $- 1$ .

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

Raise $- 1$ to the power of $2$ .

$- 4 + 3 \cdot 1 - 2 \cdot - 1 - 1$

Multiply $3$ by $1$ .

$- 4 + 3 - 2 \cdot - 1 - 1$

Add $- 4$ and $3$ .

$- 1 - 2 \cdot - 1 - 1$

Multiply $- 2$ by $- 1$ .

$- 1 + 2 - 1$

Add $- 1$ and $2$ .

$1 - 1$

Subtract $1$ from $1$ .

$0$

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.

$\frac{4 x^{3} + 3 x^{2} - 2 x - 1}{x + 1}$

Divide $4 x^{3} + 3 x^{2} - 2 x - 1$ by $x + 1$ .

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

$x$

$1$

$4 x^{3}$

$3 x^{2}$

$2 x$

$1$

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $x$ .

					$4 x^{2}$
	$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			+	$4 x^{3}$	+	$4 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 4 x^{2}$

				$4 x^{2}$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$

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

				$4 x^{2}$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$

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

				$4 x^{2}$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$

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

				$4 x^{2}$	-	$x$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$x$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					-	$x^{2}$	-	$x$

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

				$4 x^{2}$	-	$x$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					+	$x^{2}$	+	$x$

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

				$4 x^{2}$	-	$x$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					+	$x^{2}$	+	$x$
							-	$x$

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

				$4 x^{2}$	-	$x$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					+	$x^{2}$	+	$x$
							-	$x$	-	$1$

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

				$4 x^{2}$	-	$x$	-	$1$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					+	$x^{2}$	+	$x$
							-	$x$	-	$1$

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$x$	-	$1$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					+	$x^{2}$	+	$x$
							-	$x$	-	$1$
							-	$x$	-	$1$

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 1$

				$4 x^{2}$	-	$x$	-	$1$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					+	$x^{2}$	+	$x$
							-	$x$	-	$1$
							+	$x$	+	$1$

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

				$4 x^{2}$	-	$x$	-	$1$
$x$	+	$1$		$4 x^{3}$	+	$3 x^{2}$	-	$2 x$	-	$1$
			-	$4 x^{3}$	-	$4 x^{2}$
					-	$x^{2}$	-	$2 x$
					+	$x^{2}$	+	$x$
							-	$x$	-	$1$
							+	$x$	+	$1$
										$0$

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

$4 x^{2} - x - 1$

Write $4 x^{3} + 3 x^{2} - 2 x - 1$ as a set of factors.

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

Remove unnecessary parentheses.

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

Factor 32x^4+24x^3-16x^2-8x

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