Precalculus Examples

$f (x) = 2 x^{3} - 4 x^{2} - 26 x + 52$

Step 1

Set $2 x^{3} - 4 x^{2} - 26 x + 52$ equal to $0$ .

$2 x^{3} - 4 x^{2} - 26 x + 52 = 0$

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 $2$ out of $2 x^{3} - 4 x^{2} - 26 x + 52$ .

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Step 2.1.1.1

Factor $2$ out of $2 x^{3}$ .

$2 (x^{3}) - 4 x^{2} - 26 x + 52 = 0$

Step 2.1.1.2

Factor $2$ out of $- 4 x^{2}$ .

$2 (x^{3}) + 2 (- 2 x^{2}) - 26 x + 52 = 0$

Step 2.1.1.3

Factor $2$ out of $- 26 x$ .

$2 (x^{3}) + 2 (- 2 x^{2}) + 2 (- 13 x) + 52 = 0$

Step 2.1.1.4

Factor $2$ out of $52$ .

$2 x^{3} + 2 (- 2 x^{2}) + 2 (- 13 x) + 2 \cdot 26 = 0$

Step 2.1.1.5

Factor $2$ out of $2 x^{3} + 2 (- 2 x^{2})$ .

$2 (x^{3} - 2 x^{2}) + 2 (- 13 x) + 2 \cdot 26 = 0$

Step 2.1.1.6

Factor $2$ out of $2 (x^{3} - 2 x^{2}) + 2 (- 13 x)$ .

$2 (x^{3} - 2 x^{2} - 13 x) + 2 \cdot 26 = 0$

Step 2.1.1.7

Factor $2$ out of $2 (x^{3} - 2 x^{2} - 13 x) + 2 \cdot 26$ .

$2 (x^{3} - 2 x^{2} - 13 x + 26) = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$2 ((x^{3} - 2 x^{2}) - 13 x + 26) = 0$

Step 2.1.2.2

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

$2 (x^{2} (x - 2) - 13 (x - 2)) = 0$

Step 2.1.3

Factor.

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Step 2.1.3.1

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$2 ((x - 2) (x^{2} - 13)) = 0$

Step 2.1.3.2

Remove unnecessary parentheses.

$2 (x - 2) (x^{2} - 13) = 0$

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 $0$ .

$x - 2 = 0$

$x^{2} - 13 = 0$

Step 2.3

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

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Step 2.3.1

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

$x - 2 = 0$

Step 2.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.4

Set $x^{2} - 13$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - 13$ equal to $0$ .

$x^{2} - 13 = 0$

Step 2.4.2

Solve $x^{2} - 13 = 0$ for $x$ .

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Step 2.4.2.1

Add $13$ to both sides of the equation.

$x^{2} = 13$

Step 2.4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{13}$

Step 2.4.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{13}$

Step 2.4.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{13}$

Step 2.4.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{13}, - \sqrt{13}$

Step 2.5

The final solution is all the values that make $2 (x - 2) (x^{2} - 13) = 0$ true.

$x = 2, \sqrt{13}, - \sqrt{13}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 2, \sqrt{13}, - \sqrt{13}$

Decimal Form:

$x = 2, 3.60555127 \dots, - 3.60555127 \dots$

Step 4