Algebra Examples

$f (x) = 2 x^{3} + 8 x^{2} - 2 x - 8$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 x^{3} + 8 x^{2} - 2 x - 8$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{3} + 8 x^{2} - 2 x - 8 = 0$ .

$2 x^{3} + 8 x^{2} - 2 x - 8 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $2$ out of $2 x^{3} + 8 x^{2} - 2 x - 8$ .

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Step 1.2.2.1.1

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

$2 (x^{3}) + 8 x^{2} - 2 x - 8 = 0$

Step 1.2.2.1.2

Factor $2$ out of $8 x^{2}$ .

$2 (x^{3}) + 2 (4 x^{2}) - 2 x - 8 = 0$

Step 1.2.2.1.3

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

$2 (x^{3}) + 2 (4 x^{2}) + 2 (- x) - 8 = 0$

Step 1.2.2.1.4

Factor $2$ out of $- 8$ .

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

Step 1.2.2.1.5

Factor $2$ out of $2 (x^{3}) + 2 (4 x^{2})$ .

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

Step 1.2.2.1.6

Factor $2$ out of $2 (x^{3} + 4 x^{2}) + 2 (- x)$ .

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

Step 1.2.2.1.7

Factor $2$ out of $2 (x^{3} + 4 x^{2} - x) + 2 (- 4)$ .

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

Step 1.2.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2.2.2

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

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

Step 1.2.2.3

Factor the polynomial by factoring out the greatest common factor, $x + 4$ .

$2 ((x + 4) (x^{2} - 1)) = 0$

Step 1.2.2.4

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

$2 ((x + 4) (x^{2} - 1^{2})) = 0$

Step 1.2.2.5

Factor.

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Step 1.2.2.5.1

Factor.

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Step 1.2.2.5.1.1

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 $b = 1$ .

$2 ((x + 4) ((x + 1) (x - 1))) = 0$

Step 1.2.2.5.1.2

Remove unnecessary parentheses.

$2 ((x + 4) (x + 1) (x - 1)) = 0$

Step 1.2.2.5.2

Remove unnecessary parentheses.

$2 (x + 4) (x + 1) (x - 1) = 0$

Step 1.2.3

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

$x + 4 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 1.2.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 1.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 1 = 0$

Step 1.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

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

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Step 1.2.6.1

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

$x - 1 = 0$

Step 1.2.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.7

The final solution is all the values that make $2 (x + 4) (x + 1) (x - 1) = 0$ true.

$x = - 4, - 1, 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 4, 0), (- 1, 0), (1, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 {(0)}^{3} + 8 {(0)}^{2} - 2 \cdot 0 - 8$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{3} + 8 {(0)}^{2} - 2 \cdot 0 - 8$

Step 2.2.2

Remove parentheses.

$y = 2 \cdot 0^{3} + 8 \cdot 0^{2} - 2 \cdot 0 - 8$

Step 2.2.3

Remove parentheses.

$y = 2 {(0)}^{3} + 8 {(0)}^{2} - 2 \cdot 0 - 8$

Step 2.2.4

Simplify $2 {(0)}^{3} + 8 {(0)}^{2} - 2 \cdot 0 - 8$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 2 \cdot 0 + 8 {(0)}^{2} - 2 \cdot 0 - 8$

Step 2.2.4.1.2

Multiply $2$ by $0$ .

$y = 0 + 8 {(0)}^{2} - 2 \cdot 0 - 8$

Step 2.2.4.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 + 8 \cdot 0 - 2 \cdot 0 - 8$

Step 2.2.4.1.4

Multiply $8$ by $0$ .

$y = 0 + 0 - 2 \cdot 0 - 8$

Step 2.2.4.1.5

Multiply $- 2$ by $0$ .

$y = 0 + 0 + 0 - 8$

Step 2.2.4.2

Simplify by adding and subtracting.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 - 8$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 - 8$

Step 2.2.4.2.3

Subtract $8$ from $0$ .

$y = - 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 8)$

Step 3

List the intersections.

x-intercept(s): $(- 4, 0), (- 1, 0), (1, 0)$

y-intercept(s): $(0, - 8)$

Step 4