Precalculus Examples

$p (x) = x^{4} - x^{3} - 2 x - 4$

Step 1

Set $x^{4} - x^{3} - 2 x - 4$ equal to $0$ .

$x^{4} - x^{3} - 2 x - 4 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $x^{4} - x^{3} - 2 x - 4$ using the rational roots test.

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Step 2.1.1.1

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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

Step 2.1.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 2.1.1.3

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 2.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - {(- 1)}^{3} - 2 \cdot - 1 - 4$

Step 2.1.1.3.2

Raise $- 1$ to the power of $4$ .

$1 - {(- 1)}^{3} - 2 \cdot - 1 - 4$

Step 2.1.1.3.3

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

$1 - - 1 - 2 \cdot - 1 - 4$

Step 2.1.1.3.4

Multiply $- 1$ by $- 1$ .

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

Step 2.1.1.3.5

Add $1$ and $1$ .

$2 - 2 \cdot - 1 - 4$

Step 2.1.1.3.6

Multiply $- 2$ by $- 1$ .

$2 + 2 - 4$

Step 2.1.1.3.7

Add $2$ and $2$ .

$4 - 4$

Step 2.1.1.3.8

Subtract $4$ from $4$ .

$0$

Step 2.1.1.4

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.

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

Step 2.1.1.5

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

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Step 2.1.1.5.1

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

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

Step 2.1.1.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

Step 2.1.1.5.4

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
			-	$x^{4}$	-	$x^{3}$

Step 2.1.1.5.5

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
			-	$x^{4}$	-	$x^{3}$
					-	$2 x^{3}$

Step 2.1.1.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

Step 2.1.1.5.7

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$2 x^{2}$
$x$	+	$1$		$x^{4}$	-	$x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
			-	$x^{4}$	-	$x^{3}$
					-	$2 x^{3}$	+	$0 x^{2}$
					-	$2 x^{3}$	-	$2 x^{2}$

Step 2.1.1.5.9

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

Step 2.1.1.5.10

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

Step 2.1.1.5.11

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

$x^{3}$

$2 x^{2}$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

Step 2.1.1.5.12

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

$2 x^{2}$

$2 x$

Step 2.1.1.5.14

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

$2 x^{2}$

$2 x$

Step 2.1.1.5.15

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

$2 x^{2}$

$2 x$

$4 x$

Step 2.1.1.5.16

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

$2 x^{2}$

$2 x$

$4 x$

$4$

Step 2.1.1.5.17

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$2 x$

$4$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

$2 x^{2}$

$2 x$

$4 x$

$4$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$2 x^{2}$	+	$2 x$	-	$4$
$x$	+	$1$		$x^{4}$	-	$x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
			-	$x^{4}$	-	$x^{3}$
					-	$2 x^{3}$	+	$0 x^{2}$
					+	$2 x^{3}$	+	$2 x^{2}$
							+	$2 x^{2}$	-	$2 x$
							-	$2 x^{2}$	-	$2 x$
									-	$4 x$	-	$4$
									-	$4 x$	-	$4$

Step 2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 4$

$x^{3}$

$2 x^{2}$

$2 x$

$4$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

$2 x^{2}$

$2 x$

$4 x$

$4$

$4 x$

$4$

Step 2.1.1.5.20

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

$x^{3}$

$2 x^{2}$

$2 x$

$4$

$x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$4$

$x^{4}$

$x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$2 x$

$2 x^{2}$

$2 x$

$4 x$

$4$

$4 x$

$4$

$0$

Step 2.1.1.5.21

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

$x^{3} - 2 x^{2} + 2 x - 4$

Step 2.1.1.6

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

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

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.

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

Step 2.1.2.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

Remove unnecessary parentheses.

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

Step 2.2

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

$x + 1 = 0$

$x - 2 = 0$

$x^{2} + 2 = 0$

Step 2.3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4

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

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Step 2.4.1

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

$x - 2 = 0$

Step 2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.5

Set $x^{2} + 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} + 2$ equal to $0$ .

$x^{2} + 2 = 0$

Step 2.5.2

Solve $x^{2} + 2 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 2.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 2}$

Step 2.5.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 2.5.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 2.5.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 2.5.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{2}$

Step 2.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{2}$

Step 2.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{2}$

Step 2.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{2}, - i \sqrt{2}$

Step 2.6

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

$x = - 1, 2, i \sqrt{2}, - i \sqrt{2}$

Step 3