Precalculus Examples

$x^{3} + 4 x^{2} - 9 x - 36$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36$

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.

${(3)}^{3} + 4 {(3)}^{2} - 9 \cdot 3 - 36$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 3$ 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 $3$ to the power of $3$ .

$27 + 4 {(3)}^{2} - 9 \cdot 3 - 36$

Step 4.1.2

Raise $3$ to the power of $2$ .

$27 + 4 \cdot 9 - 9 \cdot 3 - 36$

Step 4.1.3

Multiply $4$ by $9$ .

$27 + 36 - 9 \cdot 3 - 36$

Step 4.1.4

Multiply $- 9$ by $3$ .

$27 + 36 - 27 - 36$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $27$ and $36$ .

$63 - 27 - 36$

Step 4.2.2

Subtract $27$ from $63$ .

$36 - 36$

Step 4.2.3

Subtract $36$ from $36$ .

$0$

Step 5

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 4 x^{2} - 9 x - 36}{x - 3}$

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.

$3$	$1$	$4$	$- 9$	$- 36$

Step 6.2

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

$3$	$1$	$4$	$- 9$	$- 36$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(4)$ .

$3$	$1$	$4$	$- 9$	$- 36$
		$3$
	$1$

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.

$3$	$1$	$4$	$- 9$	$- 36$
		$3$
	$1$	$7$

Step 6.5

Multiply the newest entry in the result $(7)$ by the divisor $(3)$ and place the result of $(21)$ under the next term in the dividend $(- 9)$ .

$3$	$1$	$4$	$- 9$	$- 36$
		$3$	$21$
	$1$	$7$

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.

$3$	$1$	$4$	$- 9$	$- 36$
		$3$	$21$
	$1$	$7$	$12$

Step 6.7

Multiply the newest entry in the result $(12)$ by the divisor $(3)$ and place the result of $(36)$ under the next term in the dividend $(- 36)$ .

$3$	$1$	$4$	$- 9$	$- 36$
		$3$	$21$	$36$
	$1$	$7$	$12$

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.

$3$	$1$	$4$	$- 9$	$- 36$
		$3$	$21$	$36$
	$1$	$7$	$12$	$0$

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.

$1 x^{2} + 7 x + 12$

Step 6.10

Simplify the quotient polynomial.

$x^{2} + 7 x + 12$

Step 7

Factor $x^{2} + 7 x + 12$ using the AC method.

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Step 7.1

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 $12$ and whose sum is $7$ .

$3, 4$

Step 7.2

Write the factored form using these integers.

$(x - 3) + (x + 3) (x + 4)$

$(x - 3) (x + 3) (x + 4)$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor out the greatest common factor from each group.

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Step 8.1.1

Group the first two terms and the last two terms.

$(x^{3} + 4 x^{2}) - 9 x - 36 = 0$

Step 8.1.2

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

$x^{2} (x + 4) - 9 (x + 4) = 0$

Step 8.2

Factor the polynomial by factoring out the greatest common factor, $x + 4$ .

$(x + 4) (x^{2} - 9) = 0$

Step 8.3

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

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

Step 8.4

Factor.

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Step 8.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$(x + 4) ((x + 3) (x - 3)) = 0$

Step 8.4.2

Remove unnecessary parentheses.

$(x + 4) (x + 3) (x - 3) = 0$

Step 9

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

$x + 4 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 10

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

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Step 10.1

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

$x + 4 = 0$

Step 10.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 11

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 11.1

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

$x + 3 = 0$

Step 11.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 12

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 12.2

Add $3$ to both sides of the equation.

$x = 3$

Step 13

The final solution is all the values that make $(x + 4) (x + 3) (x - 3) = 0$ true.

$x = - 4, - 3, 3$

Step 14