Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Simplify $\frac{- x^{3} + 2 x^{2} + 4}{x - 8}$ .

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Step 1.2.2.1

Factor $- 1$ out of $- x^{3}$ .

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

Step 1.2.2.2

Factor $- 1$ out of $2 x^{2}$ .

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

Step 1.2.2.3

Factor $- 1$ out of $- (x^{3}) - (- 2 x^{2})$ .

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

Step 1.2.2.4

Rewrite $4$ as $- 1 (- 4)$ .

$0 = \frac{- (x^{3} - 2 x^{2}) - 1 (- 4)}{x - 8}$

Step 1.2.2.5

Factor $- 1$ out of $- (x^{3} - 2 x^{2}) - 1 (- 4)$ .

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

Step 1.2.2.6

Simplify the expression.

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Step 1.2.2.6.1

Rewrite $- (x^{3} - 2 x^{2} - 4)$ as $- 1 (x^{3} - 2 x^{2} - 4)$ .

$0 = \frac{- 1 (x^{3} - 2 x^{2} - 4)}{x - 8}$

Step 1.2.2.6.2

Move the negative in front of the fraction.

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

Step 1.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 2.59431301$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2.59431301, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

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

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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 $0$ .

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

Step 2.2.5.1.2

Multiply $- 1$ by $0$ .

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

Step 2.2.5.1.3

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

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

Step 2.2.5.1.4

Multiply $2$ by $0$ .

$y = \frac{0 + 0 + 4}{0 - 8}$

Step 2.2.5.1.5

Add $0$ and $0$ .

$y = \frac{0 + 4}{0 - 8}$

Step 2.2.5.1.6

Add $0$ and $4$ .

$y = \frac{4}{0 - 8}$

Step 2.2.5.2

Reduce the expression by cancelling the common factors.

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Step 2.2.5.2.1

Subtract $8$ from $0$ .

$y = \frac{4}{- 8}$

Step 2.2.5.2.2

Cancel the common factor of $4$ and $- 8$ .

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Step 2.2.5.2.2.1

Factor $4$ out of $4$ .

$y = \frac{4 (1)}{- 8}$

Step 2.2.5.2.2.2

Cancel the common factors.

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Step 2.2.5.2.2.2.1

Factor $4$ out of $- 8$ .

$y = \frac{4 \cdot 1}{4 \cdot - 2}$

Step 2.2.5.2.2.2.2

Cancel the common factor.

$y = \frac{4 \cdot 1}{4 \cdot - 2}$

Step 2.2.5.2.2.2.3

Rewrite the expression.

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

Step 2.2.5.2.3

Move the negative in front of the fraction.

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

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(2.59431301, 0)$

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

Step 4