Algebra Examples

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

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

Set the numerator equal to zero.

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

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

Factor the left side of the equation.

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Step 2.2.1.1

Factor out the greatest common factor from each group.

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Step 2.2.1.1.1

Group the first two terms and the last two terms.

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Factor.

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Step 2.2.1.4.1

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 2.2.1.4.2

Remove unnecessary parentheses.

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

Step 2.2.2

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 2.2.3

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

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Step 2.2.3.1

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

$x - 3 = 0$

Step 2.2.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.2.4

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

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Step 2.2.4.1

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

$x + 2 = 0$

Step 2.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.2.5

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

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Step 2.2.5.1

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

$x - 2 = 0$

Step 2.2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.2.6

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

$x = 3, - 2, 2$

Step 2.3

Exclude the solutions that do not make $\frac{x^{3} - 3 x^{2} - 4 x + 12}{x - 3} = 0$ true.

$x = - 2, 2$

Step 3