Algebra Examples

$f (x) = \frac{2 x^{3} + x^{2} - 8 x - 4}{x^{2} - 3 x + 2}$

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 $x$ .

$0 = \frac{2 x^{3} + x^{2} - 8 x - 4}{x^{2} - 3 x + 2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

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

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor the left side of the equation.

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Step 1.2.2.1.1

Factor out the greatest common factor from each group.

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Step 1.2.2.1.1.1

Group the first two terms and the last two terms.

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

Step 1.2.2.1.1.2

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

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

Step 1.2.2.1.2

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 1.2.2.1.3

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

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

Step 1.2.2.1.4

Factor.

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Step 1.2.2.1.4.1

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 $b = 2$ .

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

Step 1.2.2.1.4.2

Remove unnecessary parentheses.

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

Step 1.2.2.2

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

$2 x + 1 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 1.2.2.3

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 1.2.2.3.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 1.2.2.3.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 1.2.2.3.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 1.2.2.3.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.2.3.2.2.2

Simplify the left side.

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Step 1.2.2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 1.2.2.3.2.2.3

Simplify the right side.

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Step 1.2.2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 1.2.2.4

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

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Step 1.2.2.4.1

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

$x + 2 = 0$

Step 1.2.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.2.5

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

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Step 1.2.2.5.1

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

$x - 2 = 0$

Step 1.2.2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.2.6

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

$x = - \frac{1}{2}, - 2, 2$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{2 x^{3} + x^{2} - 8 x - 4}{x^{2} - 3 x + 2}$ true.

$x = - \frac{1}{2}, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{2}, 0), (- 2, 0)$

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 $y$ .

$y = \frac{2 {(0)}^{3} + {(0)}^{2} - 8 \cdot 0 - 4}{{(0)}^{2} - 3 \cdot 0 + 2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2 \cdot 0^{3} + {(0)}^{2} - 8 \cdot 0 - 4}{{(0)}^{2} - 3 \cdot 0 + 2}$

Step 2.2.2

Remove parentheses.

$y = \frac{2 \cdot 0^{3} + 0^{2} - 8 \cdot 0 - 4}{{(0)}^{2} - 3 \cdot 0 + 2}$

Step 2.2.3

Remove parentheses.

$y = \frac{2 \cdot 0^{3} + 0^{2} - 8 \cdot 0 - 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.4

Remove parentheses.

$y = \frac{2 {(0)}^{3} + {(0)}^{2} - 8 \cdot 0 - 4}{{(0)}^{2} - 3 \cdot 0 + 2}$

Step 2.2.5

Simplify $\frac{2 {(0)}^{3} + {(0)}^{2} - 8 \cdot 0 - 4}{{(0)}^{2} - 3 \cdot 0 + 2}$ .

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

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

$y = \frac{2 \cdot 0 + 0^{2} - 8 \cdot 0 - 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.5.1.2

Multiply $2$ by $0$ .

$y = \frac{0 + 0^{2} - 8 \cdot 0 - 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.5.1.3

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

$y = \frac{0 + 0 - 8 \cdot 0 - 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.5.1.4

Multiply $- 8$ by $0$ .

$y = \frac{0 + 0 + 0 - 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.5.1.5

Add $0$ and $0$ .

$y = \frac{0 + 0 - 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.5.1.6

Add $0$ and $0$ .

$y = \frac{0 - 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.5.1.7

Subtract $4$ from $0$ .

$y = \frac{- 4}{0^{2} - 3 \cdot 0 + 2}$

Step 2.2.5.2

Simplify the denominator.

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Step 2.2.5.2.1

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

$y = \frac{- 4}{0 - 3 \cdot 0 + 2}$

Step 2.2.5.2.2

Multiply $- 3$ by $0$ .

$y = \frac{- 4}{0 + 0 + 2}$

Step 2.2.5.2.3

Add $0$ and $0$ .

$y = \frac{- 4}{0 + 2}$

Step 2.2.5.2.4

Add $0$ and $2$ .

$y = \frac{- 4}{2}$

Step 2.2.5.3

Divide $- 4$ by $2$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{2}, 0), (- 2, 0)$

y-intercept(s): $(0, - 2)$

Step 4