Algebra Examples

$y = 2 x^{3} - 7 x^{2} - 7 x + 12$

Step 1

Factor $2 x^{3} - 7 x^{2} - 7 x + 12$ 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.

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

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm 0.5, \pm 12, \pm 6, \pm 2, \pm 3, \pm 1.5, \pm 4$

Step 1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 1.3.1

Substitute $1$ into the polynomial.

$2 \cdot 1^{3} - 7 \cdot 1^{2} - 7 \cdot 1 + 12$

Step 1.3.2

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

$2 \cdot 1 - 7 \cdot 1^{2} - 7 \cdot 1 + 12$

Step 1.3.3

Multiply $2$ by $1$ .

$2 - 7 \cdot 1^{2} - 7 \cdot 1 + 12$

Step 1.3.4

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

$2 - 7 \cdot 1 - 7 \cdot 1 + 12$

Step 1.3.5

Multiply $- 7$ by $1$ .

$2 - 7 - 7 \cdot 1 + 12$

Step 1.3.6

Subtract $7$ from $2$ .

$- 5 - 7 \cdot 1 + 12$

Step 1.3.7

Multiply $- 7$ by $1$ .

$- 5 - 7 + 12$

Step 1.3.8

Subtract $7$ from $- 5$ .

$- 12 + 12$

Step 1.3.9

Add $- 12$ and $12$ .

$0$

Step 1.4

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.

$\frac{2 x^{3} - 7 x^{2} - 7 x + 12}{x - 1}$

Step 1.5

Divide $2 x^{3} - 7 x^{2} - 7 x + 12$ by $x - 1$ .

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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 $0$ .

$x$

$1$

$2 x^{3}$

$7 x^{2}$

$7 x$

$12$

Step 1.5.2

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

					$2 x^{2}$
	$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			+	$2 x^{3}$	-	$2 x^{2}$

Step 1.5.4

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$

Step 1.5.5

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$

Step 1.5.6

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$

Step 1.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					-	$5 x^{2}$	+	$5 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} + 5 x$

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					+	$5 x^{2}$	-	$5 x$

Step 1.5.10

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

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					+	$5 x^{2}$	-	$5 x$
							-	$12 x$

Step 1.5.11

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

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					+	$5 x^{2}$	-	$5 x$
							-	$12 x$	+	$12$

Step 1.5.12

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

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					+	$5 x^{2}$	-	$5 x$
							-	$12 x$	+	$12$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					+	$5 x^{2}$	-	$5 x$
							-	$12 x$	+	$12$
							-	$12 x$	+	$12$

Step 1.5.14

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

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					+	$5 x^{2}$	-	$5 x$
							-	$12 x$	+	$12$
							+	$12 x$	-	$12$

Step 1.5.15

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

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	-	$7 x$	+	$12$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	-	$7 x$
					+	$5 x^{2}$	-	$5 x$
							-	$12 x$	+	$12$
							+	$12 x$	-	$12$
										$0$

Step 1.5.16

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

$2 x^{2} - 5 x - 12$

Step 1.6

Write $2 x^{3} - 7 x^{2} - 7 x + 12$ as a set of factors.

$y = (x - 1) (2 x^{2} - 5 x - 12)$

Step 2

Factor by grouping.

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Step 2.1

Factor by grouping.

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Step 2.1.1

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 $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = - 5$ .

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Step 2.1.1.1

Factor $- 5$ out of $- 5 x$ .

$y = (x - 1) (2 x^{2} - 5 (x) - 12)$

Step 2.1.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$y = (x - 1) (2 x^{2} + (3 - 8) x - 12)$

Step 2.1.1.3

Apply the distributive property.

$y = (x - 1) (2 x^{2} + 3 x - 8 x - 12)$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$y = (x - 1) ((2 x^{2} + 3 x) - 8 x - 12)$

Step 2.1.2.2

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

$y = (x - 1) (x (2 x + 3) - 4 (2 x + 3))$

Step 2.1.3

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

$y = (x - 1) ((2 x + 3) (x - 4))$

Step 2.2

Remove unnecessary parentheses.

$y = (x - 1) (2 x + 3) (x - 4)$