Calculus Examples

$f (x) = \frac{3 x^{3} - x^{2} - 48 x + 16}{x^{2} + 5 x + 4}$

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 $x$ .

$0 = \frac{3 x^{3} - x^{2} - 48 x + 16}{x^{2} + 5 x + 4}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$3 x^{3} - x^{2} - 48 x + 16 = 0$

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.

$(3 x^{3} - x^{2}) - 48 x + 16 = 0$

Step 1.2.2.1.1.2

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

$x^{2} (3 x - 1) - 16 (3 x - 1) = 0$

Step 1.2.2.1.2

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x^{2} - 16) = 0$

Step 1.2.2.1.3

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

$(3 x - 1) ((x + 4) (x - 4)) = 0$

Step 1.2.2.1.4.2

Remove unnecessary parentheses.

$(3 x - 1) (x + 4) (x - 4) = 0$

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 $0$ .

$3 x - 1 = 0$

$x + 4 = 0$

$x - 4 = 0$

Step 1.2.2.3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 1.2.2.3.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 1.2.2.3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 1.2.2.3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 1.2.2.3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

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 $3$ .

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Step 1.2.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 1.2.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 1.2.2.4

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

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Step 1.2.2.4.1

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

$x + 4 = 0$

Step 1.2.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.2.5

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

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Step 1.2.2.5.1

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

$x - 4 = 0$

Step 1.2.2.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.2.6

The final solution is all the values that make $(3 x - 1) (x + 4) (x - 4) = 0$ true.

$x = \frac{1}{3}, - 4, 4$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{3 x^{3} - x^{2} - 48 x + 16}{x^{2} + 5 x + 4}$ true.

$x = \frac{1}{3}, 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{1}{3}, 0), (4, 0)$

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 $y$ .

$y = \frac{3 {(0)}^{3} - {(0)}^{2} - 48 \cdot 0 + 16}{{(0)}^{2} + 5 (0) + 4}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{3 \cdot 0^{3} - {(0)}^{2} - 48 \cdot 0 + 16}{{(0)}^{2} + 5 (0) + 4}$

Step 2.2.2

Remove parentheses.

$y = \frac{3 \cdot 0^{3} - 0^{2} - 48 \cdot 0 + 16}{{(0)}^{2} + 5 (0) + 4}$

Step 2.2.3

Remove parentheses.

$y = \frac{3 \cdot 0^{3} - 0^{2} - 48 \cdot 0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.4

Remove parentheses.

$y = \frac{3 {(0)}^{3} - {(0)}^{2} - 48 \cdot 0 + 16}{{(0)}^{2} + 5 (0) + 4}$

Step 2.2.5

Simplify $\frac{3 {(0)}^{3} - {(0)}^{2} - 48 \cdot 0 + 16}{{(0)}^{2} + 5 (0) + 4}$ .

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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 $0$ .

$y = \frac{3 \cdot 0 - 0^{2} - 48 \cdot 0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.5.1.2

Multiply $3$ by $0$ .

$y = \frac{0 - 0^{2} - 48 \cdot 0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.5.1.3

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

$y = \frac{0 - 0 - 48 \cdot 0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.5.1.4

Multiply $- 1$ by $0$ .

$y = \frac{0 + 0 - 48 \cdot 0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.5.1.5

Multiply $- 48$ by $0$ .

$y = \frac{0 + 0 + 0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.5.1.6

Add $0$ and $0$ .

$y = \frac{0 + 0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.5.1.7

Add $0$ and $0$ .

$y = \frac{0 + 16}{0^{2} + 5 (0) + 4}$

Step 2.2.5.1.8

Add $0$ and $16$ .

$y = \frac{16}{0^{2} + 5 (0) + 4}$

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 $0$ .

$y = \frac{16}{0 + 5 (0) + 4}$

Step 2.2.5.2.2

Multiply $5$ by $0$ .

$y = \frac{16}{0 + 0 + 4}$

Step 2.2.5.2.3

Add $0$ and $0$ .

$y = \frac{16}{0 + 4}$

Step 2.2.5.2.4

Add $0$ and $4$ .

$y = \frac{16}{4}$

Step 2.2.5.3

Divide $16$ by $4$ .

$y = 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $(\frac{1}{3}, 0), (4, 0)$

y-intercept(s): $(0, 4)$

Step 4