Factor f(x)=-x^3+13x-12

$f (x) = - x^{3} + 13 x - 12$

Step 1

Factor $- x^{3} + 13 x - 12$ 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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4 q = \pm 1$

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

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.

$- 1^{3} + 13 \cdot 1 - 12$

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

$- 1 \cdot 1 + 13 \cdot 1 - 12$

Multiply $- 1$ by $1$ .

$- 1 + 13 \cdot 1 - 12$

Multiply $13$ by $1$ .

$- 1 + 13 - 12$

Add $- 1$ and $13$ .

$12 - 12$

Subtract $12$ from $12$ .

$0$

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.

$\frac{- x^{3} + 13 x - 12}{x - 1}$

Divide $- x^{3} + 13 x - 12$ 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 $0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$13 x$

$12$

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

				-	$x^{2}$
	$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			-	$x^{3}$	+	$x^{2}$

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					-	$x^{2}$	+	$x$

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					+	$x^{2}$	-	$x$

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					+	$x^{2}$	-	$x$
							+	$12 x$

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					+	$x^{2}$	-	$x$
							+	$12 x$	-	$12$

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	-	$x$	+	$12$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					+	$x^{2}$	-	$x$
							+	$12 x$	-	$12$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$x$	+	$12$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					+	$x^{2}$	-	$x$
							+	$12 x$	-	$12$
							+	$12 x$	-	$12$

The expression needs to be subtracted from the dividend, so change all the signs in $12 x - 12$

			-	$x^{2}$	-	$x$	+	$12$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					+	$x^{2}$	-	$x$
							+	$12 x$	-	$12$
							-	$12 x$	+	$12$

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

			-	$x^{2}$	-	$x$	+	$12$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$13 x$	-	$12$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$13 x$
					+	$x^{2}$	-	$x$
							+	$12 x$	-	$12$
							-	$12 x$	+	$12$
										$0$

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

$- x^{2} - x + 12$

Write $- x^{3} + 13 x - 12$ as a set of factors.

$f (x) = (x - 1) (- x^{2} - x + 12)$

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 $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$f (x) = (x - 1) (- x^{2} - (x) + 12)$

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

$f (x) = (x - 1) (- x^{2} + (3 - 4) x + 12)$

Apply the distributive property.

$f (x) = (x - 1) (- x^{2} + 3 x - 4 x + 12)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = (x - 1) ((- x^{2} + 3 x) - 4 x + 12)$

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

$f (x) = (x - 1) (x (- x + 3) + 4 (- x + 3))$

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

$f (x) = (x - 1) ((- x + 3) (x + 4))$

Remove unnecessary parentheses.

$f (x) = (x - 1) (- x + 3) (x + 4)$

Factor f(x)=-x^3+13x-12

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