Precalculus Examples

$x^{3} + 7 x^{2} - 16 x + 18$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

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.

${(- 9)}^{3} + 7 {(- 9)}^{2} - 16 \cdot - 9 + 18$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 9$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 9$ to the power of $3$ .

$- 729 + 7 {(- 9)}^{2} - 16 \cdot - 9 + 18$

Step 4.1.2

Raise $- 9$ to the power of $2$ .

$- 729 + 7 \cdot 81 - 16 \cdot - 9 + 18$

Step 4.1.3

Multiply $7$ by $81$ .

$- 729 + 567 - 16 \cdot - 9 + 18$

Step 4.1.4

Multiply $- 16$ by $- 9$ .

$- 729 + 567 + 144 + 18$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $- 729$ and $567$ .

$- 162 + 144 + 18$

Step 4.2.2

Add $- 162$ and $144$ .

$- 18 + 18$

Step 4.2.3

Add $- 18$ and $18$ .

$0$

Step 5

Since $- 9$ is a known root, divide the polynomial by $x + 9$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 7 x^{2} - 16 x + 18}{x + 9}$

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.

$- 9$	$1$	$7$	$- 16$	$18$

Step 6.2

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

$- 9$	$1$	$7$	$- 16$	$18$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 9)$ and place the result of $(- 9)$ under the next term in the dividend $(7)$ .

$- 9$	$1$	$7$	$- 16$	$18$
		$- 9$
	$1$

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.

$- 9$	$1$	$7$	$- 16$	$18$
		$- 9$
	$1$	$- 2$

Step 6.5

Multiply the newest entry in the result $(- 2)$ by the divisor $(- 9)$ and place the result of $(18)$ under the next term in the dividend $(- 16)$ .

$- 9$	$1$	$7$	$- 16$	$18$
		$- 9$	$18$
	$1$	$- 2$

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.

$- 9$	$1$	$7$	$- 16$	$18$
		$- 9$	$18$
	$1$	$- 2$	$2$

Step 6.7

Multiply the newest entry in the result $(2)$ by the divisor $(- 9)$ and place the result of $(- 18)$ under the next term in the dividend $(18)$ .

$- 9$	$1$	$7$	$- 16$	$18$
		$- 9$	$18$	$- 18$
	$1$	$- 2$	$2$

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.

$- 9$	$1$	$7$	$- 16$	$18$
		$- 9$	$18$	$- 18$
	$1$	$- 2$	$2$	$0$

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.

$1 x^{2} + - 2 x + 2$

Step 6.10

Simplify the quotient polynomial.

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

Step 7

Solve the equation to find any remaining roots.

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Step 7.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 7.3

Simplify.

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Step 7.3.1

Simplify the numerator.

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Step 7.3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 7.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 7.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 7.3.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 7.3.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 7.3.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 7.3.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.3.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.3.1.7

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 7.3.1.8

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

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 7.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 7.3.2

Multiply $2$ by $1$ .

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

Step 7.3.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 7.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.4.1

Simplify the numerator.

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Step 7.4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 7.4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 7.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 7.4.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 7.4.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 7.4.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 7.4.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.4.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.4.1.7

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 7.4.1.8

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

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 7.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 7.4.2

Multiply $2$ by $1$ .

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

Step 7.4.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 7.4.4

Change the $\pm$ to $+$ .

$x = 1 + i$

Step 7.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.5.1

Simplify the numerator.

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Step 7.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 7.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 7.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 7.5.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 7.5.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 7.5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 7.5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.5.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.5.1.7

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 7.5.1.8

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

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 7.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 7.5.2

Multiply $2$ by $1$ .

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

Step 7.5.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 7.5.4

Change the $\pm$ to $-$ .

$x = 1 - i$

Step 7.6

The final answer is the combination of both solutions.

$x = 1 + i, 1 - i$

Step 8

The polynomial can be written as a set of linear factors.

$(x + 9) (x - 1 - i) (x - 1 + i)$

Step 9

These are the roots (zeros) of the polynomial $x^{3} + 7 x^{2} - 16 x + 18$ .

$x = - 9, 1 + i, 1 - i$

Step 10