Examples

$x^{3} - 6 x^{2} + 12 x - 8$

Step 1

Factor

$x^{3} - 6 x^{2} + 12 x - 8$ using the rational roots test.

Tap for more steps...

Step 1.1

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.

$p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1$

Step 1.2

Find every combination of

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

$\pm 1, \pm 8, \pm 2, \pm 4$

Step 1.3

Substitute

$2$ and simplify the expression. In this case, the expression is equal to

$0$ so

$2$ is a root of the polynomial.

Tap for more steps...

Step 1.3.1

Substitute

$2$ into the polynomial.

$2^{3} - 6 \cdot 2^{2} + 12 \cdot 2 - 8$

Step 1.3.2

Raise

$2$ to the power of

$3$ .

$8 - 6 \cdot 2^{2} + 12 \cdot 2 - 8$

Step 1.3.3

Raise

$2$ to the power of

$2$ .

$8 - 6 \cdot 4 + 12 \cdot 2 - 8$

Step 1.3.4

Multiply

$- 6$ by

$4$ .

$8 - 24 + 12 \cdot 2 - 8$

Step 1.3.5

Subtract

$24$ from

$8$ .

$- 16 + 12 \cdot 2 - 8$

Step 1.3.6

Multiply

$12$ by

$2$ .

$- 16 + 24 - 8$

Step 1.3.7

Add

$- 16$ and

$24$ .

$8 - 8$

Step 1.3.8

Subtract

$8$ from

$8$ .

$0$

Step 1.4

Since

$2$ is a known root, divide the polynomial by

$x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 6 x^{2} + 12 x - 8}{x - 2}$

Step 1.5

Divide

$x^{3} - 6 x^{2} + 12 x - 8$ by

$x - 2$ .

Tap for more steps...

Step 1.5.1

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

$0$ .

$x$

$2$

$x^{3}$

$6 x^{2}$

$12 x$

$8$

Step 1.5.2

Divide the highest order term in the dividend

$x^{3}$ by the highest order term in divisor

$x$ .

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			+	$x^{3}$	-	$2 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$

Step 1.5.5

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$

Step 1.5.6

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 1.5.7

Divide the highest order term in the dividend

$- 4 x^{2}$ by the highest order term in divisor

$x$ .

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					-	$4 x^{2}$	+	$8 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

$- 4 x^{2} + 8 x$

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$

Step 1.5.10

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

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$

Step 1.5.11

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

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$

Step 1.5.12

Divide the highest order term in the dividend

$4 x$ by the highest order term in divisor

$x$ .

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							+	$4 x$	-	$8$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

$4 x - 8$

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							-	$4 x$	+	$8$

Step 1.5.15

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

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							-	$4 x$	+	$8$
										$0$

Step 1.5.16

Since the remainder is

$0$ , the final answer is the quotient.

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

Step 1.6

Write

$x^{3} - 6 x^{2} + 12 x - 8$ as a set of factors.

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

Step 2

Factor using the perfect square rule.

Tap for more steps...

Step 2.1

Rewrite

$4$ as

$2^{2}$ .

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

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.3

Rewrite the polynomial.

$(x - 2) (x^{2} - 2 \cdot x \cdot 2 + 2^{2})$

Step 2.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = x$ and

$b = 2$ .

$(x - 2) {(x - 2)}^{2}$

Step 3

Combine like factors.

Tap for more steps...

Step 3.1

Raise

$x - 2$ to the power of

$1$ .

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

Step 3.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3

Add

$1$ and

$2$ .

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

Step 4

Since the polynomial can be factored, it is not prime.

Not prime

Enter YOUR Problem