Factor 2x^3-x^2-5x-2

$2 x^{3} - x^{2} - 5 x - 2$

Step 1

Factor $2 x^{3} - x^{2} - 5 x - 2$ 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.

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

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

$\pm 1, \pm 0.5, \pm 2$

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.

$2 {(- 1)}^{3} - {(- 1)}^{2} - 5 \cdot - 1 - 2$

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

$2 \cdot - 1 - {(- 1)}^{2} - 5 \cdot - 1 - 2$

Multiply $2$ by $- 1$ .

$- 2 - {(- 1)}^{2} - 5 \cdot - 1 - 2$

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

$- 2 - 1 \cdot 1 - 5 \cdot - 1 - 2$

Multiply $- 1$ by $1$ .

$- 2 - 1 - 5 \cdot - 1 - 2$

Subtract $1$ from $- 2$ .

$- 3 - 5 \cdot - 1 - 2$

Multiply $- 5$ by $- 1$ .

$- 3 + 5 - 2$

Add $- 3$ and $5$ .

$2 - 2$

Subtract $2$ from $2$ .

$0$

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.

$\frac{2 x^{3} - x^{2} - 5 x - 2}{x + 1}$

Divide $2 x^{3} - x^{2} - 5 x - 2$ 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 $0$ .

$x$

$1$

$2 x^{3}$

$x^{2}$

$5 x$

$2$

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

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			+	$2 x^{3}$	+	$2 x^{2}$

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$

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

				$2 x^{2}$	-	$3 x$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					-	$3 x^{2}$	-	$3 x$

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

				$2 x^{2}$	-	$3 x$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$3 x$

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

				$2 x^{2}$	-	$3 x$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$3 x$
							-	$2 x$

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

				$2 x^{2}$	-	$3 x$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$3 x$
							-	$2 x$	-	$2$

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

				$2 x^{2}$	-	$3 x$	-	$2$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$3 x$
							-	$2 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$	-	$2$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$3 x$
							-	$2 x$	-	$2$
							-	$2 x$	-	$2$

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

				$2 x^{2}$	-	$3 x$	-	$2$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$3 x$
							-	$2 x$	-	$2$
							+	$2 x$	+	$2$

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

				$2 x^{2}$	-	$3 x$	-	$2$
$x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$5 x$	-	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$3 x$
							-	$2 x$	-	$2$
							+	$2 x$	+	$2$
										$0$

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

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

Write $2 x^{3} - x^{2} - 5 x - 2$ as a set of factors.

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

Step 2

Factor by grouping.

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

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

Rewrite $- 3$ as $1$ plus $- 4$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Remove unnecessary parentheses.

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

Factor 2x^3-x^2-5x-2

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