Precalculus Examples

$x^{4} - 2 x^{3} + 8 x^{2} - 32 x - 128 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$- 2 x^{3} + 8 x^{2} + x^{4} - 32 x - 128 = 0$

Step 1.2

Factor $- 2 x^{2}$ out of $- 2 x^{3} + 8 x^{2}$ .

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Step 1.2.1

Factor $- 2 x^{2}$ out of $- 2 x^{3}$ .

$- 2 x^{2} x + 8 x^{2} + x^{4} - 32 x - 128 = 0$

Step 1.2.2

Factor $- 2 x^{2}$ out of $8 x^{2}$ .

$- 2 x^{2} x - 2 x^{2} \cdot - 4 + x^{4} - 32 x - 128 = 0$

Step 1.2.3

Factor $- 2 x^{2}$ out of $- 2 x^{2} (x) - 2 x^{2} (- 4)$ .

$- 2 x^{2} (x - 4) + x^{4} - 32 x - 128 = 0$

Step 1.3

Factor $x^{4} - 32 x - 128$ using the rational roots test.

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Step 1.3.1

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.

$p = \pm 1, \pm 128, \pm 2, \pm 64, \pm 4, \pm 32, \pm 8, \pm 16$

$q = \pm 1$

Step 1.3.2

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

$\pm 1, \pm 128, \pm 2, \pm 64, \pm 4, \pm 32, \pm 8, \pm 16$

Step 1.3.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 1.3.3.1

Substitute $4$ into the polynomial.

$4^{4} - 32 \cdot 4 - 128$

Step 1.3.3.2

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

$256 - 32 \cdot 4 - 128$

Step 1.3.3.3

Multiply $- 32$ by $4$ .

$256 - 128 - 128$

Step 1.3.3.4

Subtract $128$ from $256$ .

$128 - 128$

Step 1.3.3.5

Subtract $128$ from $128$ .

$0$

Step 1.3.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 32 x - 128}{x - 4}$

Step 1.3.5

Divide $x^{4} - 32 x - 128$ by $x - 4$ .

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Step 1.3.5.1

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

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

Step 1.3.5.2

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

$x^{3}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

Step 1.3.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

Step 1.3.5.4

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

$x^{3}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

Step 1.3.5.5

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

$x^{3}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

Step 1.3.5.6

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

$x^{3}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

Step 1.3.5.7

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

$x^{3}$

$4 x^{2}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

Step 1.3.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

Step 1.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 16 x^{2}$

$x^{3}$

$4 x^{2}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

Step 1.3.5.10

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

$x^{3}$

$4 x^{2}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

Step 1.3.5.11

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

$x^{3}$

$4 x^{2}$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

Step 1.3.5.12

Divide the highest order term in the dividend $16 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$16 x$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

Step 1.3.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$16 x$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

Step 1.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{2} - 64 x$

$x^{3}$

$4 x^{2}$

$16 x$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

Step 1.3.5.15

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

$x^{3}$

$4 x^{2}$

$16 x$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

$32 x$

Step 1.3.5.16

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

$x^{3}$

$4 x^{2}$

$16 x$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

$32 x$

$128$

Step 1.3.5.17

Divide the highest order term in the dividend $32 x$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$16 x$

$32$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

$32 x$

$128$

Step 1.3.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$16 x$

$32$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

$32 x$

$128$

$32 x$

$128$

Step 1.3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $32 x - 128$

$x^{3}$

$4 x^{2}$

$16 x$

$32$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

$32 x$

$128$

$32 x$

$128$

Step 1.3.5.20

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

$x^{3}$

$4 x^{2}$

$16 x$

$32$

$x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$32 x$

$128$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$16 x^{2}$

$32 x$

$16 x^{2}$

$64 x$

$32 x$

$128$

$32 x$

$128$

$0$

Step 1.3.5.21

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

$x^{3} + 4 x^{2} + 16 x + 32$

Step 1.3.6

Write $x^{4} - 32 x - 128$ as a set of factors.

$- 2 x^{2} (x - 4) + (x - 4) (x^{3} + 4 x^{2} + 16 x + 32) = 0$

Step 1.4

Factor $x - 4$ out of $- 2 x^{2} (x - 4) + (x - 4) (x^{3} + 4 x^{2} + 16 x + 32)$ .

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Step 1.4.1

Factor $x - 4$ out of $- 2 x^{2} (x - 4)$ .

$(x - 4) (- 2 x^{2}) + (x - 4) (x^{3} + 4 x^{2} + 16 x + 32) = 0$

Step 1.4.2

Factor $x - 4$ out of $(x - 4) (- 2 x^{2}) + (x - 4) (x^{3} + 4 x^{2} + 16 x + 32)$ .

$(x - 4) (- 2 x^{2} + x^{3} + 4 x^{2} + 16 x + 32) = 0$

Step 1.5

Add $- 2 x^{2}$ and $4 x^{2}$ .

$(x - 4) (x^{3} + 2 x^{2} + 16 x + 32) = 0$

Step 1.6

Factor.

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Step 1.6.1

Rewrite $x^{3} + 2 x^{2} + 16 x + 32$ in a factored form.

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Step 1.6.1.1

Factor out the greatest common factor from each group.

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Step 1.6.1.1.1

Group the first two terms and the last two terms.

$(x - 4) ((x^{3} + 2 x^{2}) + 16 x + 32) = 0$

Step 1.6.1.1.2

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

$(x - 4) (x^{2} (x + 2) + 16 (x + 2)) = 0$

Step 1.6.1.2

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

$(x - 4) ((x + 2) (x^{2} + 16)) = 0$

Step 1.6.2

Remove unnecessary parentheses.

$(x - 4) (x + 2) (x^{2} + 16) = 0$

Step 2

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

$x - 4 = 0$

$x + 2 = 0$

$x^{2} + 16 = 0$

Step 3

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

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Step 3.1

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

$x - 4 = 0$

Step 3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4

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

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Step 4.1

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

$x + 2 = 0$

Step 4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Set $x^{2} + 16$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 16$ equal to $0$ .

$x^{2} + 16 = 0$

Step 5.2

Solve $x^{2} + 16 = 0$ for $x$ .

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Step 5.2.1

Subtract $16$ from both sides of the equation.

$x^{2} = - 16$

Step 5.2.2

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

$x = \pm \sqrt{- 16}$

Step 5.2.3

Simplify $\pm \sqrt{- 16}$ .

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Step 5.2.3.1

Rewrite $- 16$ as $- 1 (16)$ .

$x = \pm \sqrt{- 1 (16)}$

Step 5.2.3.2

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{16}$

Step 5.2.3.3

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

$x = \pm i \cdot \sqrt{16}$

Step 5.2.3.4

Rewrite $16$ as $4^{2}$ .

$x = \pm i \cdot \sqrt{4^{2}}$

Step 5.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 4$

Step 5.2.3.6

Move $4$ to the left of $i$ .

$x = \pm 4 i$

Step 5.2.4

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

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Step 5.2.4.1

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

$x = 4 i$

Step 5.2.4.2

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

$x = - 4 i$

Step 5.2.4.3

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

$x = 4 i, - 4 i$

Step 6

The final solution is all the values that make $(x - 4) (x + 2) (x^{2} + 16) = 0$ true.

$x = 4, - 2, 4 i, - 4 i$

Step 7