Precalculus Examples

$f (x) = 6 x^{4} - 7 x^{3} - 36 x^{2} + 7 x + 6$

Step 1

Factor $6 x^{4} - 7 x^{3} - 36 x^{2} + 7 x + 6$ 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.

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

$q = \pm 1, \pm 6, \pm 2, \pm 3$

Step 1.2

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 6, \pm 3, \pm 2, \pm 0.\overline{6}, \pm 1.5$

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.

$6 {(- 0.\overline{3})}^{4} - 7 {(- 0.\overline{3})}^{3} - 36 {(- 0.\overline{3})}^{2} + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.2

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

$6 \cdot 0.01234567 - 7 {(- 0.\overline{3})}^{3} - 36 {(- 0.\overline{3})}^{2} + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.3

Multiply $6$ by $0.01234567$ .

$0.\overline{074} - 7 {(- 0.\overline{3})}^{3} - 36 {(- 0.\overline{3})}^{2} + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.4

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

$0.\overline{074} - 7 \cdot - 0.\overline{037} - 36 {(- 0.\overline{3})}^{2} + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.5

Multiply $- 7$ by $- 0.\overline{037}$ .

$0.\overline{074} + 0.\overline{259} - 36 {(- 0.\overline{3})}^{2} + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.6

Add $0.\overline{074}$ and $0.\overline{259}$ .

$0.\overline{3} - 36 {(- 0.\overline{3})}^{2} + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.7

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

$0.\overline{3} - 36 \cdot 0.\overline{1} + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.8

Multiply $- 36$ by $0.\overline{1}$ .

$0.\overline{3} - 4 + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.9

Subtract $4$ from $0.\overline{3}$ .

$- 3.66666666 + 7 \cdot - 0.\overline{3} + 6$

Step 1.3.10

Multiply $7$ by $- 0.\overline{3}$ .

$- 3.66666666 - 2.\overline{3} + 6$

Step 1.3.11

Subtract $2.\overline{3}$ from $- 3.66666666$ .

$- 6 + 6$

Step 1.3.12

Add $- 6$ and $6$ .

$0$

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.

$\frac{6 x^{4} - 7 x^{3} - 36 x^{2} + 7 x + 6}{3 x + 1}$

Step 1.5

Divide $6 x^{4} - 7 x^{3} - 36 x^{2} + 7 x + 6$ 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 $0$ .

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

Step 1.5.2

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

Step 1.5.3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} + 2 x^{3}$

$2 x^{3}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

Step 1.5.5

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

$2 x^{3}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

Step 1.5.6

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

$2 x^{3}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

Step 1.5.7

Divide the highest order term in the dividend $- 9 x^{3}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x^{2}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{3}$	-	$3 x^{2}$
$3 x$	+	$1$		$6 x^{4}$	-	$7 x^{3}$	-	$36 x^{2}$	+	$7 x$	+	$6$
			-	$6 x^{4}$	-	$2 x^{3}$
					-	$9 x^{3}$	-	$36 x^{2}$
					-	$9 x^{3}$	-	$3 x^{2}$

Step 1.5.9

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

$2 x^{3}$

$3 x^{2}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

Step 1.5.10

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

$2 x^{3}$

$3 x^{2}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

Step 1.5.11

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

$2 x^{3}$

$3 x^{2}$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

Step 1.5.12

Divide the highest order term in the dividend $- 33 x^{2}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x^{2}$

$11 x$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

Step 1.5.13

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x^{2}$

$11 x$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 33 x^{2} - 11 x$

$2 x^{3}$

$3 x^{2}$

$11 x$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

Step 1.5.15

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

$2 x^{3}$

$3 x^{2}$

$11 x$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

$18 x$

Step 1.5.16

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

$2 x^{3}$

$3 x^{2}$

$11 x$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

$18 x$

$6$

Step 1.5.17

Divide the highest order term in the dividend $18 x$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x^{2}$

$11 x$

$6$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

$18 x$

$6$

Step 1.5.18

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x^{2}$

$11 x$

$6$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

$18 x$

$6$

$18 x$

$6$

Step 1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $18 x + 6$

$2 x^{3}$

$3 x^{2}$

$11 x$

$6$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

$18 x$

$6$

$18 x$

$6$

Step 1.5.20

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

$2 x^{3}$

$3 x^{2}$

$11 x$

$6$

$3 x$

$1$

$6 x^{4}$

$7 x^{3}$

$36 x^{2}$

$7 x$

$6$

$6 x^{4}$

$2 x^{3}$

$9 x^{3}$

$36 x^{2}$

$9 x^{3}$

$3 x^{2}$

$33 x^{2}$

$7 x$

$33 x^{2}$

$11 x$

$18 x$

$6$

$18 x$

$6$

$0$

Step 1.5.21

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

$2 x^{3} - 3 x^{2} - 11 x + 6$

Step 1.6

Write $6 x^{4} - 7 x^{3} - 36 x^{2} + 7 x + 6$ as a set of factors.

$f (x) = (3 x + 1) (2 x^{3} - 3 x^{2} - 11 x + 6)$

Step 2

Factor $2 x^{3} - 3 x^{2} - 11 x + 6$ 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.

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

$q = \pm 1, \pm 2$

Step 2.2

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

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

Step 2.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 2.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{3} - 3 \cdot {0.5}^{2} - 11 \cdot 0.5 + 6$

Step 2.3.2

Raise $0.5$ to the power of $3$ .

$2 \cdot 0.125 - 3 \cdot {0.5}^{2} - 11 \cdot 0.5 + 6$

Step 2.3.3

Multiply $2$ by $0.125$ .

$0.25 - 3 \cdot {0.5}^{2} - 11 \cdot 0.5 + 6$

Step 2.3.4

Raise $0.5$ to the power of $2$ .

$0.25 - 3 \cdot 0.25 - 11 \cdot 0.5 + 6$

Step 2.3.5

Multiply $- 3$ by $0.25$ .

$0.25 - 0.75 - 11 \cdot 0.5 + 6$

Step 2.3.6

Subtract $0.75$ from $0.25$ .

$- 0.5 - 11 \cdot 0.5 + 6$

Step 2.3.7

Multiply $- 11$ by $0.5$ .

$- 0.5 - 5.5 + 6$

Step 2.3.8

Subtract $5.5$ from $- 0.5$ .

$- 6 + 6$

Step 2.3.9

Add $- 6$ and $6$ .

$0$

Step 2.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 3 x^{2} - 11 x + 6}{2 x - 1}$

Step 2.5

Divide $2 x^{3} - 3 x^{2} - 11 x + 6$ by $2 x - 1$ .

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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 $0$ .

$2 x$

$1$

$2 x^{3}$

$3 x^{2}$

$11 x$

$6$

Step 2.5.2

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

					$x^{2}$
	$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$

Step 2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			+	$2 x^{3}$	-	$x^{2}$

Step 2.5.4

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$

Step 2.5.5

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$

Step 2.5.6

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$

Step 2.5.7

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$

Step 2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					-	$2 x^{2}$	+	$x$

Step 2.5.9

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$

Step 2.5.10

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$

Step 2.5.11

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$

Step 2.5.12

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$

Step 2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$
							-	$12 x$	+	$6$

Step 2.5.14

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

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$
							+	$12 x$	-	$6$

Step 2.5.15

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

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$
							+	$12 x$	-	$6$
										$0$

Step 2.5.16

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

$x^{2} - x - 6$

Step 2.6

Write $2 x^{3} - 3 x^{2} - 11 x + 6$ as a set of factors.

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

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

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

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Step 3.1.1.1

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 3.1.1.2

Write the factored form using these integers.

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

Step 3.1.2

Remove unnecessary parentheses.

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

Step 3.2

Remove unnecessary parentheses.

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