Precalculus Examples

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

Step 1

Set $x^{3} + 3 x^{2} - 4 x - 12$ equal to $0$ .

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

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 out the greatest common factor from each group.

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Step 2.1.1.1

Group the first two terms and the last two terms.

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Factor.

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Step 2.1.4.1

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 2.1.4.2

Remove unnecessary parentheses.

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

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 $0$ .

$x + 3 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 2.3

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

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Step 2.3.1

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

$x + 3 = 0$

Step 2.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.4

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

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Step 2.4.1

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

$x + 2 = 0$

Step 2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.5

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

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Step 2.5.1

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

$x - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.6

The final solution is all the values that make $(x + 3) (x + 2) (x - 2) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 3$ (Multiplicity of $1$ )

$x = - 2$ (Multiplicity of $1$ )

$x = 2$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $1$ )

$x = - 2$ (Multiplicity of $1$ )

$x = 2$ (Multiplicity of $1$ )

Step 3