Precalculus Examples

$f (x) = x^{3} + 7 x^{2} + 15 x + 9$

Step 1

Set $x^{3} + 7 x^{2} + 15 x + 9$ equal to $0$ .

$x^{3} + 7 x^{2} + 15 x + 9 = 0$

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 $x^{3} + 7 x^{2} + 15 x + 9$ 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.

$p = \pm 1, \pm 9, \pm 3$

$q = \pm 1$

Step 2.1.1.2

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

$\pm 1, \pm 9, \pm 3$

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.

${(- 1)}^{3} + 7 {(- 1)}^{2} + 15 \cdot - 1 + 9$

Step 2.1.1.3.2

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

$- 1 + 7 {(- 1)}^{2} + 15 \cdot - 1 + 9$

Step 2.1.1.3.3

Raise $- 1$ to the power of $2$ .

$- 1 + 7 \cdot 1 + 15 \cdot - 1 + 9$

Step 2.1.1.3.4

Multiply $7$ by $1$ .

$- 1 + 7 + 15 \cdot - 1 + 9$

Step 2.1.1.3.5

Add $- 1$ and $7$ .

$6 + 15 \cdot - 1 + 9$

Step 2.1.1.3.6

Multiply $15$ by $- 1$ .

$6 - 15 + 9$

Step 2.1.1.3.7

Subtract $15$ from $6$ .

$- 9 + 9$

Step 2.1.1.3.8

Add $- 9$ and $9$ .

$0$

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.

$\frac{x^{3} + 7 x^{2} + 15 x + 9}{x + 1}$

Step 2.1.1.5

Divide $x^{3} + 7 x^{2} + 15 x + 9$ 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 $0$ .

$x$

$1$

$x^{3}$

$7 x^{2}$

$15 x$

$9$

Step 2.1.1.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			+	$x^{3}$	+	$x^{2}$

Step 2.1.1.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$

Step 2.1.1.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$

Step 2.1.1.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$

Step 2.1.1.5.7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					+	$6 x^{2}$	+	$6 x$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 6 x$

				$x^{2}$	+	$6 x$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					-	$6 x^{2}$	-	$6 x$

Step 2.1.1.5.10

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

				$x^{2}$	+	$6 x$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					-	$6 x^{2}$	-	$6 x$
							+	$9 x$

Step 2.1.1.5.11

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

				$x^{2}$	+	$6 x$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					-	$6 x^{2}$	-	$6 x$
							+	$9 x$	+	$9$

Step 2.1.1.5.12

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$	+	$9$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					-	$6 x^{2}$	-	$6 x$
							+	$9 x$	+	$9$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	+	$9$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					-	$6 x^{2}$	-	$6 x$
							+	$9 x$	+	$9$
							+	$9 x$	+	$9$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $9 x + 9$

				$x^{2}$	+	$6 x$	+	$9$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					-	$6 x^{2}$	-	$6 x$
							+	$9 x$	+	$9$
							-	$9 x$	-	$9$

Step 2.1.1.5.15

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

				$x^{2}$	+	$6 x$	+	$9$
$x$	+	$1$		$x^{3}$	+	$7 x^{2}$	+	$15 x$	+	$9$
			-	$x^{3}$	-	$x^{2}$
					+	$6 x^{2}$	+	$15 x$
					-	$6 x^{2}$	-	$6 x$
							+	$9 x$	+	$9$
							-	$9 x$	-	$9$
										$0$

Step 2.1.1.5.16

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

$x^{2} + 6 x + 9$

Step 2.1.1.6

Write $x^{3} + 7 x^{2} + 15 x + 9$ as a set of factors.

$(x + 1) (x^{2} + 6 x + 9) = 0$

Step 2.1.2

Factor using the perfect square rule.

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Step 2.1.2.1

Rewrite $9$ as $3^{2}$ .

$(x + 1) (x^{2} + 6 x + 3^{2}) = 0$

Step 2.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 2.1.2.3

Rewrite the polynomial.

$(x + 1) (x^{2} + 2 \cdot x \cdot 3 + 3^{2}) = 0$

Step 2.1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

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 $0$ .

$x + 1 = 0$

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

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 ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 2.4.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 2.4.2.1

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

$x + 3 = 0$

Step 2.4.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.5

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

$x = - 1, - 3$

Step 3