Precalculus Examples

$x^{4} + x^{3} + 14 x^{2} + 16 x - 32$

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.

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

$q = \pm 1$

Step 2

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

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

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.

${(1)}^{4} + {(1)}^{3} + 14 {(1)}^{2} + 16 (1) - 32$

Step 4

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

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

$1 + {(1)}^{3} + 14 {(1)}^{2} + 16 (1) - 32$

Step 4.1.2

One to any power is one.

$1 + 1 + 14 {(1)}^{2} + 16 (1) - 32$

Step 4.1.3

One to any power is one.

$1 + 1 + 14 \cdot 1 + 16 (1) - 32$

Step 4.1.4

Multiply $14$ by $1$ .

$1 + 1 + 14 + 16 (1) - 32$

Step 4.1.5

Multiply $16$ by $1$ .

$1 + 1 + 14 + 16 - 32$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $1$ and $1$ .

$2 + 14 + 16 - 32$

Step 4.2.2

Add $2$ and $14$ .

$16 + 16 - 32$

Step 4.2.3

Add $16$ and $16$ .

$32 - 32$

Step 4.2.4

Subtract $32$ from $32$ .

$0$

Step 5

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.

$\frac{x^{4} + x^{3} + 14 x^{2} + 16 x - 32}{x - 1}$

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.

$1$	$1$	$1$	$14$	$16$	$- 32$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$1$	$1$	$1$	$14$	$16$	$- 32$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(1)$ .

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$
	$1$

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.

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$
	$1$	$2$

Step 6.5

Multiply the newest entry in the result $(2)$ by the divisor $(1)$ and place the result of $(2)$ under the next term in the dividend $(14)$ .

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$	$2$
	$1$	$2$

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.

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$	$2$
	$1$	$2$	$16$

Step 6.7

Multiply the newest entry in the result $(16)$ by the divisor $(1)$ and place the result of $(16)$ under the next term in the dividend $(16)$ .

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$	$2$	$16$
	$1$	$2$	$16$

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.

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$	$2$	$16$
	$1$	$2$	$16$	$32$

Step 6.9

Multiply the newest entry in the result $(32)$ by the divisor $(1)$ and place the result of $(32)$ under the next term in the dividend $(- 32)$ .

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$	$2$	$16$	$32$
	$1$	$2$	$16$	$32$

Step 6.10

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

$1$	$1$	$1$	$14$	$16$	$- 32$
		$1$	$2$	$16$	$32$
	$1$	$2$	$16$	$32$	$0$

Step 6.11

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

$1 x^{3} + 2 x^{2} + (16) x + 32$

Step 6.12

Simplify the quotient polynomial.

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

Step 7

Factor out the greatest common factor from each group.

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Step 7.1

Group the first two terms and the last two terms.

$(x - 1) (x^{3} + 2 x^{2}) + 16 x + 32$

Step 7.2

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

$(x - 1) + x^{2} (x + 2) + 16 (x + 2)$

$(x - 1) x^{2} (x + 2) + 16 (x + 2)$

Step 8

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

$(x - 1) (x + 2) (x^{2} + 16)$

Step 9

Factor the left side of the equation.

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Step 9.1

Regroup terms.

$x^{3} + 16 x + x^{4} + 14 x^{2} - 32 = 0$

Step 9.2

Factor $x$ out of $x^{3} + 16 x$ .

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Step 9.2.1

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

$x \cdot x^{2} + 16 x + x^{4} + 14 x^{2} - 32 = 0$

Step 9.2.2

Factor $x$ out of $16 x$ .

$x \cdot x^{2} + x \cdot 16 + x^{4} + 14 x^{2} - 32 = 0$

Step 9.2.3

Factor $x$ out of $x \cdot x^{2} + x \cdot 16$ .

$x (x^{2} + 16) + x^{4} + 14 x^{2} - 32 = 0$

Step 9.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (x^{2} + 16) + {(x^{2})}^{2} + 14 x^{2} - 32 = 0$

Step 9.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (x^{2} + 16) + u^{2} + 14 u - 32 = 0$

Step 9.5

Factor $u^{2} + 14 u - 32$ using the AC method.

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Step 9.5.1

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 $- 32$ and whose sum is $14$ .

$- 2, 16$

Step 9.5.2

Write the factored form using these integers.

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

Step 9.6

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 9.7

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

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Step 9.7.1

Factor $x^{2} + 16$ out of $x (x^{2} + 16)$ .

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

Step 9.7.2

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

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

Step 9.7.3

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

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

Step 9.8

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 9.9

Factor $u + u^{2} - 2$ using the AC method.

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Step 9.9.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 9.9.2

Write the factored form using these integers.

$(x^{2} + 16) ((u - 1) (u + 2)) = 0$

Step 9.10

Factor.

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Step 9.10.1

Replace all occurrences of $u$ with $x$ .

$(x^{2} + 16) ((x - 1) (x + 2)) = 0$

Step 9.10.2

Remove unnecessary parentheses.

$(x^{2} + 16) (x - 1) (x + 2) = 0$

Step 10

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

$x^{2} + 16 = 0$

$x - 1 = 0$

$x + 2 = 0$

Step 11

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

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Step 11.1

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

$x^{2} + 16 = 0$

Step 11.2

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

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Step 11.2.1

Subtract $16$ from both sides of the equation.

$x^{2} = - 16$

Step 11.2.2

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

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

Step 11.2.3

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

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Step 11.2.3.1

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

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

Step 11.2.3.2

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

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

Step 11.2.3.3

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

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

Step 11.2.3.4

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

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

Step 11.2.3.5

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

$x = \pm i \cdot 4$

Step 11.2.3.6

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

$x = \pm 4 i$

Step 11.2.4

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

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Step 11.2.4.1

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

$x = 4 i$

Step 11.2.4.2

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

$x = - 4 i$

Step 11.2.4.3

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

$x = 4 i, - 4 i$

Step 12

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

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Step 12.1

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

$x - 1 = 0$

Step 12.2

Add $1$ to both sides of the equation.

$x = 1$

Step 13

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

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Step 13.1

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

$x + 2 = 0$

Step 13.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 14

The final solution is all the values that make $(x^{2} + 16) (x - 1) (x + 2) = 0$ true.

$x = 4 i, - 4 i, 1, - 2$

Step 15