Precalculus Examples

$f (x) = - 4 {(x - 4)}^{2} (x^{2} - 4)$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 4 {(x - 4)}^{2} (x^{2} - 4) = 0$ .

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

Step 1.2.2

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

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

$x^{2} - 4 = 0$

Step 1.2.3

Set ${(x - 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set ${(x - 4)}^{2}$ equal to $0$ .

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

Step 1.2.3.2

Solve ${(x - 4)}^{2} = 0$ for $x$ .

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Step 1.2.3.2.1

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

$x - 4 = 0$

Step 1.2.3.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.4

Set $x^{2} - 4$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} - 4$ equal to $0$ .

$x^{2} - 4 = 0$

Step 1.2.4.2

Solve $x^{2} - 4 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 1.2.4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 1.2.4.2.3

Simplify $\pm \sqrt{4}$ .

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Step 1.2.4.2.3.1

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

$x = \pm \sqrt{2^{2}}$

Step 1.2.4.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 1.2.4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 1.2.4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 1.2.4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 1.2.5

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

$x = 4, 2, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4, 0), (2, 0), (- 2, 0)$

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 $y$ .

$y = - 4 {((0) - 4)}^{2} ({(0)}^{2} - 4)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 4 {(0 - 4)}^{2} ({(0)}^{2} - 4)$

Step 2.2.2

Remove parentheses.

$y = - 4 {(0 - 4)}^{2} (0^{2} - 4)$

Step 2.2.3

Remove parentheses.

$y = - 4 {((0) - 4)}^{2} ({(0)}^{2} - 4)$

Step 2.2.4

Simplify $- 4 {((0) - 4)}^{2} ({(0)}^{2} - 4)$ .

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Step 2.2.4.1

Subtract $4$ from $0$ .

$y = - 4 {(- 4)}^{2} ({(0)}^{2} - 4)$

Step 2.2.4.2

Multiply $- 4$ by ${(- 4)}^{2}$ by adding the exponents.

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Step 2.2.4.2.1

Multiply $- 4$ by ${(- 4)}^{2}$ .

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Step 2.2.4.2.1.1

Raise $- 4$ to the power of $1$ .

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

Step 2.2.4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.4.2.2

Add $1$ and $2$ .

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

Step 2.2.4.3

Raise $- 4$ to the power of $3$ .

$y = - 64 ({(0)}^{2} - 4)$

Step 2.2.4.4

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

$y = - 64 (0 - 4)$

Step 2.2.4.5

Subtract $4$ from $0$ .

$y = - 64 \cdot - 4$

Step 2.2.4.6

Multiply $- 64$ by $- 4$ .

$y = 256$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 256)$

Step 3

List the intersections.

x-intercept(s): $(4, 0), (2, 0), (- 2, 0)$

y-intercept(s): $(0, 256)$

Step 4