Factor 6x^3-13x^2-4x+15

$6 x^{3} - 13 x^{2} - 4 x + 15$

Step 1

Factor $6 x^{3} - 13 x^{2} - 4 x + 15$ 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.

$p = \pm 1, \pm 15, \pm 3, \pm 5 q = \pm 1, \pm 6, \pm 2, \pm 3$

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 15, \pm 2.5, \pm 7.5, \pm 5, \pm 3, \pm 1.5, \pm 0.8\overline{3}, \pm 1.\overline{6}$

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.

$6 {(- 1)}^{3} - 13 {(- 1)}^{2} - 4 \cdot - 1 + 15$

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

$6 \cdot - 1 - 13 {(- 1)}^{2} - 4 \cdot - 1 + 15$

Multiply $6$ by $- 1$ .

$- 6 - 13 {(- 1)}^{2} - 4 \cdot - 1 + 15$

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

$- 6 - 13 \cdot 1 - 4 \cdot - 1 + 15$

Multiply $- 13$ by $1$ .

$- 6 - 13 - 4 \cdot - 1 + 15$

Subtract $13$ from $- 6$ .

$- 19 - 4 \cdot - 1 + 15$

Multiply $- 4$ by $- 1$ .

$- 19 + 4 + 15$

Add $- 19$ and $4$ .

$- 15 + 15$

Add $- 15$ and $15$ .

$0$

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.

$\frac{6 x^{3} - 13 x^{2} - 4 x + 15}{x + 1}$

Divide $6 x^{3} - 13 x^{2} - 4 x + 15$ 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 $0$ .

$x$

$1$

$6 x^{3}$

$13 x^{2}$

$4 x$

$15$

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

					$6 x^{2}$
	$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$

Multiply the new quotient term by the divisor.

				$6 x^{2}$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			+	$6 x^{3}$	+	$6 x^{2}$

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

				$6 x^{2}$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$

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

				$6 x^{2}$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$

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

				$6 x^{2}$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$

Divide the highest order term in the dividend $- 19 x^{2}$ by the highest order term in divisor $x$ .

				$6 x^{2}$	-	$19 x$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$

Multiply the new quotient term by the divisor.

				$6 x^{2}$	-	$19 x$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					-	$19 x^{2}$	-	$19 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 19 x^{2} - 19 x$

				$6 x^{2}$	-	$19 x$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					+	$19 x^{2}$	+	$19 x$

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

				$6 x^{2}$	-	$19 x$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					+	$19 x^{2}$	+	$19 x$
							+	$15 x$

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

				$6 x^{2}$	-	$19 x$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					+	$19 x^{2}$	+	$19 x$
							+	$15 x$	+	$15$

Divide the highest order term in the dividend $15 x$ by the highest order term in divisor $x$ .

				$6 x^{2}$	-	$19 x$	+	$15$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					+	$19 x^{2}$	+	$19 x$
							+	$15 x$	+	$15$

Multiply the new quotient term by the divisor.

				$6 x^{2}$	-	$19 x$	+	$15$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					+	$19 x^{2}$	+	$19 x$
							+	$15 x$	+	$15$
							+	$15 x$	+	$15$

The expression needs to be subtracted from the dividend, so change all the signs in $15 x + 15$

				$6 x^{2}$	-	$19 x$	+	$15$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					+	$19 x^{2}$	+	$19 x$
							+	$15 x$	+	$15$
							-	$15 x$	-	$15$

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

				$6 x^{2}$	-	$19 x$	+	$15$
$x$	+	$1$		$6 x^{3}$	-	$13 x^{2}$	-	$4 x$	+	$15$
			-	$6 x^{3}$	-	$6 x^{2}$
					-	$19 x^{2}$	-	$4 x$
					+	$19 x^{2}$	+	$19 x$
							+	$15 x$	+	$15$
							-	$15 x$	-	$15$
										$0$

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

$6 x^{2} - 19 x + 15$

Write $6 x^{3} - 13 x^{2} - 4 x + 15$ as a set of factors.

$(x + 1) (6 x^{2} - 19 x + 15)$

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 $a \cdot c = 6 \cdot 15 = 90$ and whose sum is $b = - 19$ .

Factor $- 19$ out of $- 19 x$ .

$(x + 1) (6 x^{2} - 19 (x) + 15)$

Rewrite $- 19$ as $- 9$ plus $- 10$

$(x + 1) (6 x^{2} + (- 9 - 10) x + 15)$

Apply the distributive property.

$(x + 1) (6 x^{2} - 9 x - 10 x + 15)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 1) ((6 x^{2} - 9 x) - 10 x + 15)$

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

$(x + 1) (3 x (2 x - 3) - 5 (2 x - 3))$

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$(x + 1) ((2 x - 3) (3 x - 5))$

Remove unnecessary parentheses.

$(x + 1) (2 x - 3) (3 x - 5)$

Factor 6x^3-13x^2-4x+15

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