Factor 6x^3+11x^2-3x-2

$6 x^{3} + 11 x^{2} - 3 x - 2$

Step 1

Factor $6 x^{3} + 11 x^{2} - 3 x - 2$ 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.

$p = \pm 1, \pm 2 q = \pm 1, \pm 6, \pm 2, \pm 3$

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 2, \pm 0.\overline{6}$

Substitute $- 0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{3}$ is a root of the polynomial.

Substitute $- 0.\overline{3}$ into the polynomial.

$6 {(- 0.\overline{3})}^{3} + 11 {(- 0.\overline{3})}^{2} - 3 \cdot - 0.\overline{3} - 2$

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

$6 \cdot - 0.\overline{037} + 11 {(- 0.\overline{3})}^{2} - 3 \cdot - 0.\overline{3} - 2$

Multiply $6$ by $- 0.\overline{037}$ .

$- 0.\overline{2} + 11 {(- 0.\overline{3})}^{2} - 3 \cdot - 0.\overline{3} - 2$

Raise $- 0.\overline{3}$ to the power of $2$ .

$- 0.\overline{2} + 11 \cdot 0.\overline{1} - 3 \cdot - 0.\overline{3} - 2$

Multiply $11$ by $0.\overline{1}$ .

$- 0.\overline{2} + 1.\overline{2} - 3 \cdot - 0.\overline{3} - 2$

Add $- 0.\overline{2}$ and $1.\overline{2}$ .

$1 - 3 \cdot - 0.\overline{3} - 2$

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

$1 + 1 - 2$

Add $1$ and $1$ .

$2 - 2$

Subtract $2$ from $2$ .

$0$

Since $- 0.\overline{3}$ is a known root, divide the polynomial by $3 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{3} + 11 x^{2} - 3 x - 2}{3 x + 1}$

Divide $6 x^{3} + 11 x^{2} - 3 x - 2$ by $3 x + 1$ .

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

$3 x$

$1$

$6 x^{3}$

$11 x^{2}$

$3 x$

$2$

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

					$2 x^{2}$
	$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			+	$6 x^{3}$	+	$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} + 2 x^{2}$

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$

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

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$

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

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $3 x$ .

				$2 x^{2}$	+	$3 x$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$3 x$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					+	$9 x^{2}$	+	$3 x$

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} + 3 x$

				$2 x^{2}$	+	$3 x$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					-	$9 x^{2}$	-	$3 x$

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

				$2 x^{2}$	+	$3 x$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					-	$9 x^{2}$	-	$3 x$
							-	$6 x$

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

				$2 x^{2}$	+	$3 x$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					-	$9 x^{2}$	-	$3 x$
							-	$6 x$	-	$2$

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $3 x$ .

				$2 x^{2}$	+	$3 x$	-	$2$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					-	$9 x^{2}$	-	$3 x$
							-	$6 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$3 x$	-	$2$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					-	$9 x^{2}$	-	$3 x$
							-	$6 x$	-	$2$
							-	$6 x$	-	$2$

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

				$2 x^{2}$	+	$3 x$	-	$2$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					-	$9 x^{2}$	-	$3 x$
							-	$6 x$	-	$2$
							+	$6 x$	+	$2$

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

				$2 x^{2}$	+	$3 x$	-	$2$
$3 x$	+	$1$		$6 x^{3}$	+	$11 x^{2}$	-	$3 x$	-	$2$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$9 x^{2}$	-	$3 x$
					-	$9 x^{2}$	-	$3 x$
							-	$6 x$	-	$2$
							+	$6 x$	+	$2$
										$0$

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

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

Write $6 x^{3} + 11 x^{2} - 3 x - 2$ as a set of factors.

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

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 $3 x$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$(3 x + 1) (x (2 x - 1) + 2 (2 x - 1))$

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

$(3 x + 1) ((2 x - 1) (x + 2))$

Remove unnecessary parentheses.

$(3 x + 1) (2 x - 1) (x + 2)$

Factor 6x^3+11x^2-3x-2

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