Precalculus Examples

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

Step 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.

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

$q = \pm 1$

Step 2

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

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

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(2)}^{3} - 6 {(2)}^{2} + 12 (2) - 8$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ is a root of the polynomial.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Raise $2$ to the power of $3$ .

$8 - 6 {(2)}^{2} + 12 (2) - 8$

Step 4.1.2

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

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

Step 4.1.3

Multiply $- 6$ by $4$ .

$8 - 24 + 12 (2) - 8$

Step 4.1.4

Multiply $12$ by $2$ .

$8 - 24 + 24 - 8$

Step 4.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.1

Subtract $24$ from $8$ .

$- 16 + 24 - 8$

Step 4.2.2

Add $- 16$ and $24$ .

$8 - 8$

Step 4.2.3

Subtract $8$ from $8$ .

$0$

Step 5

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 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Tap for more steps...

Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$2$	$1$	$- 6$	$12$	$- 8$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$2$	$1$	$- 6$	$12$	$- 8$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(2)$ and place the result of $(2)$ under the next term in the dividend $(- 6)$ .

$2$	$1$	$- 6$	$12$	$- 8$
		$2$
	$1$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$- 6$	$12$	$- 8$
		$2$
	$1$	$- 4$

Step 6.5

Multiply the newest entry in the result $(- 4)$ by the divisor $(2)$ and place the result of $(- 8)$ under the next term in the dividend $(12)$ .

$2$	$1$	$- 6$	$12$	$- 8$
		$2$	$- 8$
	$1$	$- 4$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$- 6$	$12$	$- 8$
		$2$	$- 8$
	$1$	$- 4$	$4$

Step 6.7

Multiply the newest entry in the result $(4)$ by the divisor $(2)$ and place the result of $(8)$ under the next term in the dividend $(- 8)$ .

$2$	$1$	$- 6$	$12$	$- 8$
		$2$	$- 8$	$8$
	$1$	$- 4$	$4$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$- 6$	$12$	$- 8$
		$2$	$- 8$	$8$
	$1$	$- 4$	$4$	$0$

Step 6.9

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

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

Step 6.10

Simplify the quotient polynomial.

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

Step 7

Factor using the perfect square rule.

Tap for more steps...

Step 7.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.2

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 7.3

Rewrite the polynomial.

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

Step 7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

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

Step 8

Factor the left side of the equation.

Tap for more steps...

Step 8.1

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

Tap for more steps...

Step 8.1.1

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 8.1.2

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 8.1.3

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 8.1.3.1

Substitute $2$ into the polynomial.

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

Step 8.1.3.2

Raise $2$ to the power of $3$ .

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

Step 8.1.3.3

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

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

Step 8.1.3.4

Multiply $- 6$ by $4$ .

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

Step 8.1.3.5

Subtract $24$ from $8$ .

$- 16 + 12 \cdot 2 - 8$

Step 8.1.3.6

Multiply $12$ by $2$ .

$- 16 + 24 - 8$

Step 8.1.3.7

Add $- 16$ and $24$ .

$8 - 8$

Step 8.1.3.8

Subtract $8$ from $8$ .

$0$

Step 8.1.4

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 8.1.5

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

Tap for more steps...

Step 8.1.5.1

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 8.1.5.2

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 8.1.5.3

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 8.1.5.4

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 8.1.5.5

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 8.1.5.6

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 8.1.5.7

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 8.1.5.8

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 8.1.5.9

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 8.1.5.10

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 8.1.5.11

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 8.1.5.12

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 8.1.5.13

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 8.1.5.14

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 8.1.5.15

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 8.1.5.16

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

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

Step 8.1.6

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

$(x - 2) (x^{2} - 4 x + 4) = 0$

Step 8.2

Factor using the perfect square rule.

Tap for more steps...

Step 8.2.1

Rewrite $4$ as $2^{2}$ .

$(x - 2) (x^{2} - 4 x + 2^{2}) = 0$

Step 8.2.2

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 8.2.3

Rewrite the polynomial.

$(x - 2) (x^{2} - 2 \cdot x \cdot 2 + 2^{2}) = 0$

Step 8.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$(x - 2) {(x - 2)}^{2} = 0$

Step 8.3

Combine like factors.

Tap for more steps...

Step 8.3.1

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

$(x - 2) {(x - 2)}^{2} = 0$

Step 8.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(x - 2)}^{1 + 2} = 0$

Step 8.3.3

Add $1$ and $2$ .

${(x - 2)}^{3} = 0$

Step 9

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 10

Add $2$ to both sides of the equation.

$x = 2$

Step 11