Precalculus Examples

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

Step 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.

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

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm 4, \pm 6, \pm 12$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$2 {(\frac{1}{2})}^{3} - 3 {(\frac{1}{2})}^{2} - 23 (\frac{1}{2}) + 12$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{1}{2}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to $\frac{1}{2}$ .

$2 \frac{1^{3}}{2^{3}} - 3 {(\frac{1}{2})}^{2} - 23 (\frac{1}{2}) + 12$

Step 4.1.2

One to any power is one.

$2 \frac{1}{2^{3}} - 3 {(\frac{1}{2})}^{2} - 23 (\frac{1}{2}) + 12$

Step 4.1.3

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

$2 (\frac{1}{8}) - 3 {(\frac{1}{2})}^{2} - 23 (\frac{1}{2}) + 12$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Factor $2$ out of $8$ .

$2 \frac{1}{2 (4)} - 3 {(\frac{1}{2})}^{2} - 23 (\frac{1}{2}) + 12$

Step 4.1.4.2

Cancel the common factor.

$2 \frac{1}{2 \cdot 4} - 3 {(\frac{1}{2})}^{2} - 23 (\frac{1}{2}) + 12$

Step 4.1.4.3

Rewrite the expression.

$\frac{1}{4} - 3 {(\frac{1}{2})}^{2} - 23 (\frac{1}{2}) + 12$

Step 4.1.5

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{4} - 3 \frac{1^{2}}{2^{2}} - 23 (\frac{1}{2}) + 12$

Step 4.1.6

One to any power is one.

$\frac{1}{4} - 3 \frac{1}{2^{2}} - 23 (\frac{1}{2}) + 12$

Step 4.1.7

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

$\frac{1}{4} - 3 (\frac{1}{4}) - 23 (\frac{1}{2}) + 12$

Step 4.1.8

Combine $- 3$ and $\frac{1}{4}$ .

$\frac{1}{4} + \frac{- 3}{4} - 23 (\frac{1}{2}) + 12$

Step 4.1.9

Move the negative in front of the fraction.

$\frac{1}{4} - \frac{3}{4} - 23 (\frac{1}{2}) + 12$

Step 4.1.10

Combine $- 23$ and $\frac{1}{2}$ .

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

Step 4.1.11

Move the negative in front of the fraction.

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

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Subtract $3$ from $1$ .

$12 + \frac{- 2}{4} + \frac{- 23}{2}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $12$ as a fraction with denominator $1$ .

$\frac{12}{1} + \frac{- 2}{4} + \frac{- 23}{2}$

Step 4.3.2

Multiply $\frac{12}{1}$ by $\frac{4}{4}$ .

$\frac{12}{1} \cdot \frac{4}{4} + \frac{- 2}{4} + \frac{- 23}{2}$

Step 4.3.3

Multiply $\frac{12}{1}$ by $\frac{4}{4}$ .

$\frac{12 \cdot 4}{4} + \frac{- 2}{4} + \frac{- 23}{2}$

Step 4.3.4

Multiply $\frac{- 23}{2}$ by $\frac{2}{2}$ .

$\frac{12 \cdot 4}{4} + \frac{- 2}{4} + \frac{- 23}{2} \cdot \frac{2}{2}$

Step 4.3.5

Multiply $\frac{- 23}{2}$ by $\frac{2}{2}$ .

$\frac{12 \cdot 4}{4} + \frac{- 2}{4} + \frac{- 23 \cdot 2}{2 \cdot 2}$

Step 4.3.6

Multiply $2$ by $2$ .

$\frac{12 \cdot 4}{4} + \frac{- 2}{4} + \frac{- 23 \cdot 2}{4}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{12 \cdot 4 - 2 - 23 \cdot 2}{4}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $12$ by $4$ .

$\frac{48 - 2 - 23 \cdot 2}{4}$

Step 4.5.2

Multiply $- 23$ by $2$ .

$\frac{48 - 2 - 46}{4}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Subtract $2$ from $48$ .

$\frac{46 - 46}{4}$

Step 4.6.2

Subtract $46$ from $46$ .

$\frac{0}{4}$

Step 4.6.3

Divide $0$ by $4$ .

$0$

Step 5

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

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

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(\frac{1}{2})$ and place the result of $(1)$ under the next term in the dividend $(- 3)$ .

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$
		$1$
	$2$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$
		$1$
	$2$	$- 2$

Step 6.5

Multiply the newest entry in the result $(- 2)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 1)$ under the next term in the dividend $(- 23)$ .

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$
		$1$	$- 1$
	$2$	$- 2$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$
		$1$	$- 1$
	$2$	$- 2$	$- 24$

Step 6.7

Multiply the newest entry in the result $(- 24)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 12)$ under the next term in the dividend $(12)$ .

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$
		$1$	$- 1$	$- 12$
	$2$	$- 2$	$- 24$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{1}{2}$	$2$	$- 3$	$- 23$	$12$
		$1$	$- 1$	$- 12$
	$2$	$- 2$	$- 24$	$0$

Step 6.9

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

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

Step 6.10

Simplify the quotient polynomial.

$2 x^{2} - 2 x - 24$

Step 7

Factor $2$ out of $2 x^{2} - 2 x - 24$ .

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Step 7.1

Factor $2$ out of $2 x^{2}$ .

$(x - \frac{1}{2}) (2 (x^{2}) - 2 x - 24)$

Step 7.2

Factor $2$ out of $- 2 x$ .

$(x - \frac{1}{2}) (2 (x^{2}) + 2 (- x) - 24)$

Step 7.3

Factor $2$ out of $- 24$ .

$(x - \frac{1}{2}) (2 (x^{2}) + 2 (- x) + 2 (- 12))$

Step 7.4

Factor $2$ out of $2 (x^{2}) + 2 (- x)$ .

$(x - \frac{1}{2}) (2 (x^{2} - x) + 2 (- 12))$

Step 7.5

Factor $2$ out of $2 (x^{2} - x) + 2 (- 12)$ .

$(x - \frac{1}{2}) (2 (x^{2} - x - 12))$

Step 8

Factor.

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Step 8.1

Factor $x^{2} - x - 12$ using the AC method.

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Step 8.1.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 8.1.2

Write the factored form using these integers.

$(x - \frac{1}{2}) (2 ((x - 4) (x + 3)))$

Step 8.2

Remove unnecessary parentheses.

$(x - \frac{1}{2}) (2 (x - 4) (x + 3))$

Step 9

Factor the left side of the equation.

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Step 9.1

Factor $2 x^{3} - 3 x^{2} - 23 x + 12$ using the rational roots test.

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Step 9.1.1

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 9.1.2

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 9.1.3

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 9.1.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{3} - 3 \cdot {0.5}^{2} - 23 \cdot 0.5 + 12$

Step 9.1.3.2

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

$2 \cdot 0.125 - 3 \cdot {0.5}^{2} - 23 \cdot 0.5 + 12$

Step 9.1.3.3

Multiply $2$ by $0.125$ .

$0.25 - 3 \cdot {0.5}^{2} - 23 \cdot 0.5 + 12$

Step 9.1.3.4

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

$0.25 - 3 \cdot 0.25 - 23 \cdot 0.5 + 12$

Step 9.1.3.5

Multiply $- 3$ by $0.25$ .

$0.25 - 0.75 - 23 \cdot 0.5 + 12$

Step 9.1.3.6

Subtract $0.75$ from $0.25$ .

$- 0.5 - 23 \cdot 0.5 + 12$

Step 9.1.3.7

Multiply $- 23$ by $0.5$ .

$- 0.5 - 11.5 + 12$

Step 9.1.3.8

Subtract $11.5$ from $- 0.5$ .

$- 12 + 12$

Step 9.1.3.9

Add $- 12$ and $12$ .

$0$

Step 9.1.4

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.

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

Step 9.1.5

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

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Step 9.1.5.1

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

$2 x$

$1$

$2 x^{3}$

$3 x^{2}$

$23 x$

$12$

Step 9.1.5.2

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

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

Step 9.1.5.3

Multiply the new quotient term by the divisor.

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

Step 9.1.5.4

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

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

Step 9.1.5.5

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

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

Step 9.1.5.6

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

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

Step 9.1.5.7

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

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

Step 9.1.5.8

Multiply the new quotient term by the divisor.

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

Step 9.1.5.9

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

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

Step 9.1.5.10

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

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

Step 9.1.5.11

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

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

Step 9.1.5.12

Divide the highest order term in the dividend $- 24 x$ by the highest order term in divisor $2 x$ .

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

Step 9.1.5.13

Multiply the new quotient term by the divisor.

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

Step 9.1.5.14

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

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

Step 9.1.5.15

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

				$x^{2}$	-	$x$	-	$12$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$23 x$	+	$12$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$23 x$
					+	$2 x^{2}$	-	$x$
							-	$24 x$	+	$12$
							+	$24 x$	-	$12$
										$0$

Step 9.1.5.16

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

$x^{2} - x - 12$

Step 9.1.6

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

$(2 x - 1) (x^{2} - x - 12) = 0$

Step 9.2

Factor $x^{2} - x - 12$ using the AC method.

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Step 9.2.1

Factor $x^{2} - x - 12$ using the AC method.

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Step 9.2.1.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 9.2.1.2

Write the factored form using these integers.

$(2 x - 1) ((x - 4) (x + 3)) = 0$

Step 9.2.2

Remove unnecessary parentheses.

$(2 x - 1) (x - 4) (x + 3) = 0$

Step 10

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 1 = 0$

$x - 4 = 0$

$x + 3 = 0$

Step 11

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 11.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 11.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 11.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 11.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 11.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 12

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 12.2

Add $4$ to both sides of the equation.

$x = 4$

Step 13

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 13.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 14

The final solution is all the values that make $(2 x - 1) (x - 4) (x + 3) = 0$ true.

$x = \frac{1}{2}, 4, - 3$

Step 15