Precalculus Examples

$3 x^{4} + 4 x^{3} - 23 x^{2} - 44 x - 12$

Step 1

Factor $3 x^{4} + 4 x^{3} - 23 x^{2} - 44 x - 12$ 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.

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

$q = \pm 1, \pm 3$

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Substitute $- 0.\overline{3}$ into the polynomial.

$3 {(- 0.\overline{3})}^{4} + 4 {(- 0.\overline{3})}^{3} - 23 {(- 0.\overline{3})}^{2} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.2

Raise $- 0.\overline{3}$ to the power of $4$ .

$3 \cdot 0.01234567 + 4 {(- 0.\overline{3})}^{3} - 23 {(- 0.\overline{3})}^{2} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.3

Multiply $3$ by $0.01234567$ .

$0.\overline{037} + 4 {(- 0.\overline{3})}^{3} - 23 {(- 0.\overline{3})}^{2} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.4

Raise $- 0.\overline{3}$ to the power of $3$ .

$0.\overline{037} + 4 \cdot - 0.\overline{037} - 23 {(- 0.\overline{3})}^{2} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.5

Multiply $4$ by $- 0.\overline{037}$ .

$0.\overline{037} - 0.\overline{148} - 23 {(- 0.\overline{3})}^{2} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.6

Subtract $0.\overline{148}$ from $0.\overline{037}$ .

$- 0.11111111 - 23 {(- 0.\overline{3})}^{2} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.7

Raise $- 0.\overline{3}$ to the power of $2$ .

$- 0.11111111 - 23 \cdot 0.\overline{1} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.8

Multiply $- 23$ by $0.\overline{1}$ .

$- 0.11111111 - 2.\overline{5} - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.9

Subtract $2.\overline{5}$ from $- 0.11111111$ .

$- 2.66666666 - 44 \cdot - 0.\overline{3} - 12$

Step 1.3.10

Multiply $- 44$ by $- 0.\overline{3}$ .

$- 2.66666666 + 14.\overline{6} - 12$

Step 1.3.11

Add $- 2.66666666$ and $14.\overline{6}$ .

$12 - 12$

Step 1.3.12

Subtract $12$ from $12$ .

$0$

Step 1.4

Since $- 0.\overline{3}$ is a known root, divide the polynomial by $3 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{4} + 4 x^{3} - 23 x^{2} - 44 x - 12}{3 x + 1}$

Step 1.5

Divide $3 x^{4} + 4 x^{3} - 23 x^{2} - 44 x - 12$ by $3 x + 1$ .

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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 $0$ .

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

Step 1.5.2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

Step 1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + x^{3}$

				$x^{3}$
$3 x$	+	$1$		$3 x^{4}$	+	$4 x^{3}$	-	$23 x^{2}$	-	$44 x$	-	$12$
			-	$3 x^{4}$	-	$x^{3}$

Step 1.5.5

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

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

Step 1.5.6

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

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

Step 1.5.7

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

$x^{3}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

Step 1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

Step 1.5.9

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

$x^{3}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

Step 1.5.10

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

$x^{3}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

$24 x^{2}$

Step 1.5.11

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

$x^{3}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

$24 x^{2}$

$44 x$

Step 1.5.12

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$x^{2}$

$8 x$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

$24 x^{2}$

$44 x$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$x^{2}$	-	$8 x$
$3 x$	+	$1$		$3 x^{4}$	+	$4 x^{3}$	-	$23 x^{2}$	-	$44 x$	-	$12$
			-	$3 x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$23 x^{2}$
					-	$3 x^{3}$	-	$x^{2}$
							-	$24 x^{2}$	-	$44 x$
							-	$24 x^{2}$	-	$8 x$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} - 8 x$

$x^{3}$

$x^{2}$

$8 x$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

$24 x^{2}$

$44 x$

$24 x^{2}$

$8 x$

Step 1.5.15

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

$x^{3}$

$x^{2}$

$8 x$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

$24 x^{2}$

$44 x$

$24 x^{2}$

$8 x$

$36 x$

Step 1.5.16

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

$x^{3}$

$x^{2}$

$8 x$

$3 x$

$1$

$3 x^{4}$

$4 x^{3}$

$23 x^{2}$

$44 x$

$12$

$3 x^{4}$

$x^{3}$

$3 x^{3}$

$23 x^{2}$

$3 x^{3}$

$x^{2}$

$24 x^{2}$

$44 x$

$24 x^{2}$

$8 x$

$36 x$

$12$

Step 1.5.17

Divide the highest order term in the dividend $- 36 x$ by the highest order term in divisor $3 x$ .

				$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
$3 x$	+	$1$		$3 x^{4}$	+	$4 x^{3}$	-	$23 x^{2}$	-	$44 x$	-	$12$
			-	$3 x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$23 x^{2}$
					-	$3 x^{3}$	-	$x^{2}$
							-	$24 x^{2}$	-	$44 x$
							+	$24 x^{2}$	+	$8 x$
									-	$36 x$	-	$12$

Step 1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
$3 x$	+	$1$		$3 x^{4}$	+	$4 x^{3}$	-	$23 x^{2}$	-	$44 x$	-	$12$
			-	$3 x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$23 x^{2}$
					-	$3 x^{3}$	-	$x^{2}$
							-	$24 x^{2}$	-	$44 x$
							+	$24 x^{2}$	+	$8 x$
									-	$36 x$	-	$12$
									-	$36 x$	-	$12$

Step 1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 36 x - 12$

				$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
$3 x$	+	$1$		$3 x^{4}$	+	$4 x^{3}$	-	$23 x^{2}$	-	$44 x$	-	$12$
			-	$3 x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$23 x^{2}$
					-	$3 x^{3}$	-	$x^{2}$
							-	$24 x^{2}$	-	$44 x$
							+	$24 x^{2}$	+	$8 x$
									-	$36 x$	-	$12$
									+	$36 x$	+	$12$

Step 1.5.20

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

				$x^{3}$	+	$x^{2}$	-	$8 x$	-	$12$
$3 x$	+	$1$		$3 x^{4}$	+	$4 x^{3}$	-	$23 x^{2}$	-	$44 x$	-	$12$
			-	$3 x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$23 x^{2}$
					-	$3 x^{3}$	-	$x^{2}$
							-	$24 x^{2}$	-	$44 x$
							+	$24 x^{2}$	+	$8 x$
									-	$36 x$	-	$12$
									+	$36 x$	+	$12$
												$0$

Step 1.5.21

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

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

Step 1.6

Write $3 x^{4} + 4 x^{3} - 23 x^{2} - 44 x - 12$ as a set of factors.

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

Step 2

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

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Step 2.1

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.

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

$q = \pm 1$

Step 2.2

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

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

Step 2.3

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 2.3.1

Substitute $- 2$ into the polynomial.

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

Step 2.3.2

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

$- 8 + {(- 2)}^{2} - 8 \cdot - 2 - 12$

Step 2.3.3

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

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

Step 2.3.4

Add $- 8$ and $4$ .

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

Step 2.3.5

Multiply $- 8$ by $- 2$ .

$- 4 + 16 - 12$

Step 2.3.6

Add $- 4$ and $16$ .

$12 - 12$

Step 2.3.7

Subtract $12$ from $12$ .

$0$

Step 2.4

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.

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

Step 2.5

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

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Step 2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{3}$

$x^{2}$

$8 x$

$12$

Step 2.5.2

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

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

Step 2.5.3

Multiply the new quotient term by the divisor.

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

Step 2.5.4

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

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

Step 2.5.5

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

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

Step 2.5.6

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

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

Step 2.5.7

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

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

Step 2.5.8

Multiply the new quotient term by the divisor.

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

Step 2.5.9

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

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

Step 2.5.10

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

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

Step 2.5.11

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

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

Step 2.5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

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

Step 2.5.13

Multiply the new quotient term by the divisor.

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

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 12$

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

Step 2.5.15

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

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

Step 2.5.16

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

$x^{2} - x - 6$

Step 2.6

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

$(3 x + 1) ((x + 2) (x^{2} - x - 6))$

Step 3

Factor $x^{2} - x - 6$ using the AC method.

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Step 3.1

Factor $x^{2} - x - 6$ using the AC method.

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Step 3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 3.1.2

Write the factored form using these integers.

$(3 x + 1) ((x + 2) ((x - 3) (x + 2)))$

Step 3.2

Remove unnecessary parentheses.

$(3 x + 1) ((x + 2) (x - 3) (x + 2))$

Step 4

Combine exponents.

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Step 4.1

Combine exponents.

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Step 4.1.1

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

$(3 x + 1) ({(x + 2)}^{1} (x + 2) (x - 3))$

Step 4.1.2

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

$(3 x + 1) ({(x + 2)}^{1} {(x + 2)}^{1} (x - 3))$

Step 4.1.3

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

$(3 x + 1) ({(x + 2)}^{1 + 1} (x - 3))$

Step 4.1.4

Add $1$ and $1$ .

$(3 x + 1) ({(x + 2)}^{2} (x - 3))$

Step 4.2

Remove unnecessary parentheses.

$(3 x + 1) {(x + 2)}^{2} (x - 3)$