Trigonometry Examples

$x^{3} + 6 x^{2} - x - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$(x^{3} + 6 x^{2}) - x - 6 = 0$

Step 1.1.2

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

$x^{2} (x + 6) - (x + 6) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $x + 6$ .

$(x + 6) (x^{2} - 1) = 0$

Step 1.3

Rewrite $1$ as $1^{2}$ .

$(x + 6) (x^{2} - 1^{2}) = 0$

Step 1.4

Factor.

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Step 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 $b = 1$ .

$(x + 6) ((x + 1) (x - 1)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$(x + 6) (x + 1) (x - 1) = 0$

Step 2

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

$x + 6 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 3

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 3.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 6

The final solution is all the values that make $(x + 6) (x + 1) (x - 1) = 0$ true.

$x = - 6, - 1, 1$