Calculus Examples

$3 x^{3} - x^{2} - 22 x + 24$

Step 1

Factor $3 x^{3} - x^{2} - 22 x + 24$ 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.

$p = \pm 1, \pm 24, \pm 2, \pm 12, \pm 3, \pm 8, \pm 4, \pm 6$

$q = \pm 1, \pm 3$

Step 1.2

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

$\pm 1, \pm 0.\overline{3}, \pm 24, \pm 8, \pm 2, \pm 0.\overline{6}, \pm 12, \pm 4, \pm 3, \pm 2.\overline{6}, \pm 1.\overline{3}, \pm 6$

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.

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Step 1.3.1

Substitute $2$ into the polynomial.

$3 \cdot 2^{3} - 2^{2} - 22 \cdot 2 + 24$

Step 1.3.2

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

$3 \cdot 8 - 2^{2} - 22 \cdot 2 + 24$

Step 1.3.3

Multiply $3$ by $8$ .

$24 - 2^{2} - 22 \cdot 2 + 24$

Step 1.3.4

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

$24 - 1 \cdot 4 - 22 \cdot 2 + 24$

Step 1.3.5

Multiply $- 1$ by $4$ .

$24 - 4 - 22 \cdot 2 + 24$

Step 1.3.6

Subtract $4$ from $24$ .

$20 - 22 \cdot 2 + 24$

Step 1.3.7

Multiply $- 22$ by $2$ .

$20 - 44 + 24$

Step 1.3.8

Subtract $44$ from $20$ .

$- 24 + 24$

Step 1.3.9

Add $- 24$ and $24$ .

$0$

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.

$\frac{3 x^{3} - x^{2} - 22 x + 24}{x - 2}$

Step 1.5

Divide $3 x^{3} - x^{2} - 22 x + 24$ by $x - 2$ .

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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 $0$ .

$x$

$2$

$3 x^{3}$

$x^{2}$

$22 x$

$24$

Step 1.5.2

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

					$3 x^{2}$
	$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			+	$3 x^{3}$	-	$6 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 6 x^{2}$

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$

Step 1.5.5

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

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$

Step 1.5.6

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

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$

Step 1.5.7

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

				$3 x^{2}$	+	$5 x$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$5 x$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	-	$10 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} - 10 x$

				$3 x^{2}$	+	$5 x$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	+	$10 x$

Step 1.5.10

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

				$3 x^{2}$	+	$5 x$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	+	$10 x$
							-	$12 x$

Step 1.5.11

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

				$3 x^{2}$	+	$5 x$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	+	$10 x$
							-	$12 x$	+	$24$

Step 1.5.12

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

				$3 x^{2}$	+	$5 x$	-	$12$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	+	$10 x$
							-	$12 x$	+	$24$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$5 x$	-	$12$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	+	$10 x$
							-	$12 x$	+	$24$
							-	$12 x$	+	$24$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x + 24$

				$3 x^{2}$	+	$5 x$	-	$12$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	+	$10 x$
							-	$12 x$	+	$24$
							+	$12 x$	-	$24$

Step 1.5.15

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

				$3 x^{2}$	+	$5 x$	-	$12$
$x$	-	$2$		$3 x^{3}$	-	$x^{2}$	-	$22 x$	+	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					+	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	+	$10 x$
							-	$12 x$	+	$24$
							+	$12 x$	-	$24$
										$0$

Step 1.5.16

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

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

Step 1.6

Write $3 x^{3} - x^{2} - 22 x + 24$ as a set of factors.

$(x - 2) (3 x^{2} + 5 x - 12)$

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 $a \cdot c = 3 \cdot - 12 = - 36$ and whose sum is $b = 5$ .

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Step 2.1.1.1

Factor $5$ out of $5 x$ .

$(x - 2) (3 x^{2} + 5 (x) - 12)$

Step 2.1.1.2

Rewrite $5$ as $- 4$ plus $9$

$(x - 2) (3 x^{2} + (- 4 + 9) x - 12)$

Step 2.1.1.3

Apply the distributive property.

$(x - 2) (3 x^{2} - 4 x + 9 x - 12)$

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.

$(x - 2) ((3 x^{2} - 4 x) + 9 x - 12)$

Step 2.1.2.2

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

$(x - 2) (x (3 x - 4) + 3 (3 x - 4))$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 4$ .

$(x - 2) ((3 x - 4) (x + 3))$

Step 2.2

Remove unnecessary parentheses.

$(x - 2) (3 x - 4) (x + 3)$