Precalculus Examples

$x^{3} - 3 x^{2} - 4 x + 12$

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.

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

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

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.

${(2)}^{3} - 3 {(2)}^{2} - 4 \cdot 2 + 12$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ 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 $2$ to the power of $3$ .

$8 - 3 {(2)}^{2} - 4 \cdot 2 + 12$

Step 4.1.2

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

$8 - 3 \cdot 4 - 4 \cdot 2 + 12$

Step 4.1.3

Multiply $- 3$ by $4$ .

$8 - 12 - 4 \cdot 2 + 12$

Step 4.1.4

Multiply $- 4$ by $2$ .

$8 - 12 - 8 + 12$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $12$ from $8$ .

$- 4 - 8 + 12$

Step 4.2.2

Subtract $8$ from $- 4$ .

$- 12 + 12$

Step 4.2.3

Add $- 12$ and $12$ .

$0$

Step 5

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 3 x^{2} - 4 x + 12}{x - 2}$

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.

$2$	$1$	$- 3$	$- 4$	$12$

Step 6.2

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

$2$	$1$	$- 3$	$- 4$	$12$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(2)$ and place the result of $(2)$ under the next term in the dividend $(- 3)$ .

$2$	$1$	$- 3$	$- 4$	$12$
		$2$
	$1$

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.

$2$	$1$	$- 3$	$- 4$	$12$
		$2$
	$1$	$- 1$

Step 6.5

Multiply the newest entry in the result $(- 1)$ by the divisor $(2)$ and place the result of $(- 2)$ under the next term in the dividend $(- 4)$ .

$2$	$1$	$- 3$	$- 4$	$12$
		$2$	$- 2$
	$1$	$- 1$

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.

$2$	$1$	$- 3$	$- 4$	$12$
		$2$	$- 2$
	$1$	$- 1$	$- 6$

Step 6.7

Multiply the newest entry in the result $(- 6)$ by the divisor $(2)$ and place the result of $(- 12)$ under the next term in the dividend $(12)$ .

$2$	$1$	$- 3$	$- 4$	$12$
		$2$	$- 2$	$- 12$
	$1$	$- 1$	$- 6$

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.

$2$	$1$	$- 3$	$- 4$	$12$
		$2$	$- 2$	$- 12$
	$1$	$- 1$	$- 6$	$0$

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.

$1 x^{2} + - 1 x - 6$

Step 6.10

Simplify the quotient polynomial.

$x^{2} - x - 6$

Step 7

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 7.2

Write the factored form using these integers.

$(x - 2) + (x - 3) (x + 2)$

$(x - 2) (x - 3) (x + 2)$

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.

$(x^{3} - 3 x^{2}) - 4 x + 12 = 0$

Step 8.1.2

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

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

Step 8.2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

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

Step 8.3

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

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

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 $b = 2$ .

$(x - 3) ((x + 2) (x - 2)) = 0$

Step 8.4.2

Remove unnecessary parentheses.

$(x - 3) (x + 2) (x - 2) = 0$

Step 9

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

$x - 3 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 10

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

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Step 10.1

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

$x - 3 = 0$

Step 10.2

Add $3$ to both sides of the equation.

$x = 3$

Step 11

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

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Step 11.1

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

$x + 2 = 0$

Step 11.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 12

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

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Step 12.1

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

$x - 2 = 0$

Step 12.2

Add $2$ to both sides of the equation.

$x = 2$

Step 13

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

$x = 3, - 2, 2$

Step 14