Precalculus Examples

$x^{3} - 6 x^{2} + 12 x - 9$

Step 1

Factor $x^{3} - 6 x^{2} + 12 x - 9$ using the rational roots test.

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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.

$p = \pm 1, \pm 9, \pm 3$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 9, \pm 3$

Step 1.3

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

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Step 1.3.1

Substitute $3$ into the polynomial.

$3^{3} - 6 \cdot 3^{2} + 12 \cdot 3 - 9$

Step 1.3.2

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

$27 - 6 \cdot 3^{2} + 12 \cdot 3 - 9$

Step 1.3.3

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

$27 - 6 \cdot 9 + 12 \cdot 3 - 9$

Step 1.3.4

Multiply $- 6$ by $9$ .

$27 - 54 + 12 \cdot 3 - 9$

Step 1.3.5

Subtract $54$ from $27$ .

$- 27 + 12 \cdot 3 - 9$

Step 1.3.6

Multiply $12$ by $3$ .

$- 27 + 36 - 9$

Step 1.3.7

Add $- 27$ and $36$ .

$9 - 9$

Step 1.3.8

Subtract $9$ from $9$ .

$0$

Step 1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 6 x^{2} + 12 x - 9}{x - 3}$

Step 1.5

Divide $x^{3} - 6 x^{2} + 12 x - 9$ by $x - 3$ .

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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 $0$ .

$x$

$3$

$x^{3}$

$6 x^{2}$

$12 x$

$9$

Step 1.5.2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			+	$x^{3}$	-	$3 x^{2}$

Step 1.5.4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$

Step 1.5.5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$

Step 1.5.6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$

Step 1.5.7

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

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					-	$3 x^{2}$	+	$9 x$

Step 1.5.9

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

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					+	$3 x^{2}$	-	$9 x$

Step 1.5.10

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

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					+	$3 x^{2}$	-	$9 x$
							+	$3 x$

Step 1.5.11

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

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					+	$3 x^{2}$	-	$9 x$
							+	$3 x$	-	$9$

Step 1.5.12

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

				$x^{2}$	-	$3 x$	+	$3$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					+	$3 x^{2}$	-	$9 x$
							+	$3 x$	-	$9$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	+	$3$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					+	$3 x^{2}$	-	$9 x$
							+	$3 x$	-	$9$
							+	$3 x$	-	$9$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x - 9$

				$x^{2}$	-	$3 x$	+	$3$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					+	$3 x^{2}$	-	$9 x$
							+	$3 x$	-	$9$
							-	$3 x$	+	$9$

Step 1.5.15

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

				$x^{2}$	-	$3 x$	+	$3$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$9$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$12 x$
					+	$3 x^{2}$	-	$9 x$
							+	$3 x$	-	$9$
							-	$3 x$	+	$9$
										$0$

Step 1.5.16

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

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

Step 1.6

Write $x^{3} - 6 x^{2} + 12 x - 9$ as a set of factors.

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

Step 2

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

Not prime

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