Algebra Examples

$f (x) = - \frac{1}{3} x {(x^{2} - 25)}^{2}$

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 $x$ .

$0 = - \frac{1}{3} \cdot (x {(x^{2} - 25)}^{2})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{3} \cdot (x {(x^{2} - 25)}^{2}) = 0$ .

$- \frac{1}{3} \cdot (x {(x^{2} - 25)}^{2}) = 0$

Step 1.2.2

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{1}{3} \cdot (x {(x^{2} - 25)}^{2})) = - 3 \cdot 0$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $- 3 (- \frac{1}{3} \cdot (x {(x^{2} - 25)}^{2}))$ .

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Step 1.2.3.1.1.1

Multiply $- \frac{1}{3} (x {(x^{2} - 25)}^{2})$ .

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Step 1.2.3.1.1.1.1

Combine $x$ and $\frac{1}{3}$ .

$- 3 (- \frac{x}{3} {(x^{2} - 25)}^{2}) = - 3 \cdot 0$

Step 1.2.3.1.1.1.2

Combine ${(x^{2} - 25)}^{2}$ and $\frac{x}{3}$ .

$- 3 (- \frac{{(x^{2} - 25)}^{2} x}{3}) = - 3 \cdot 0$

Step 1.2.3.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.2.1

Move the leading negative in $- \frac{{(x^{2} - 25)}^{2} x}{3}$ into the numerator.

$- 3 \frac{- {(x^{2} - 25)}^{2} x}{3} = - 3 \cdot 0$

Step 1.2.3.1.1.2.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- {(x^{2} - 25)}^{2} x}{3} = - 3 \cdot 0$

Step 1.2.3.1.1.2.3

Cancel the common factor.

$3 \cdot - 1 \frac{- {(x^{2} - 25)}^{2} x}{3} = - 3 \cdot 0$

Step 1.2.3.1.1.2.4

Rewrite the expression.

$- (- {(x^{2} - 25)}^{2} x) = - 3 \cdot 0$

Step 1.2.3.1.1.3

Simplify the expression.

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Step 1.2.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 ({(x^{2} - 25)}^{2} x) = - 3 \cdot 0$

Step 1.2.3.1.1.3.2

Multiply ${(x^{2} - 25)}^{2}$ by $1$ .

${(x^{2} - 25)}^{2} x = - 3 \cdot 0$

Step 1.2.3.1.1.3.3

Reorder factors in ${(x^{2} - 25)}^{2} x$ .

$x {(x^{2} - 25)}^{2} = - 3 \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $- 3$ by $0$ .

$x {(x^{2} - 25)}^{2} = 0$

Step 1.2.4

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

$x = 0$

${(x^{2} - 25)}^{2} = 0$

Step 1.2.5

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.6

Set ${(x^{2} - 25)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set ${(x^{2} - 25)}^{2}$ equal to $0$ .

${(x^{2} - 25)}^{2} = 0$

Step 1.2.6.2

Solve ${(x^{2} - 25)}^{2} = 0$ for $x$ .

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Step 1.2.6.2.1

Factor the left side of the equation.

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Step 1.2.6.2.1.1

Rewrite $25$ as $5^{2}$ .

${(x^{2} - 5^{2})}^{2} = 0$

Step 1.2.6.2.1.2

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 $b = 5$ .

${((x + 5) (x - 5))}^{2} = 0$

Step 1.2.6.2.1.3

Apply the product rule to $(x + 5) (x - 5)$ .

${(x + 5)}^{2} {(x - 5)}^{2} = 0$

Step 1.2.6.2.2

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

${(x + 5)}^{2} = 0$

${(x - 5)}^{2} = 0$

Step 1.2.6.2.3

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.2.3.1

Set ${(x + 5)}^{2}$ equal to $0$ .

${(x + 5)}^{2} = 0$

Step 1.2.6.2.3.2

Solve ${(x + 5)}^{2} = 0$ for $x$ .

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Step 1.2.6.2.3.2.1

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 1.2.6.2.3.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.6.2.4

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.2.4.1

Set ${(x - 5)}^{2}$ equal to $0$ .

${(x - 5)}^{2} = 0$

Step 1.2.6.2.4.2

Solve ${(x - 5)}^{2} = 0$ for $x$ .

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Step 1.2.6.2.4.2.1

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 1.2.6.2.4.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.6.2.5

The final solution is all the values that make ${(x + 5)}^{2} {(x - 5)}^{2} = 0$ true.

$x = - 5, 5$

Step 1.2.7

The final solution is all the values that make $x {(x^{2} - 25)}^{2} = 0$ true.

$x = 0, - 5, 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 5, 0), (5, 0)$

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 $y$ .

$y = - \frac{1}{3} \cdot ((0) {({(0)}^{2} - 25)}^{2})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{1}{3} \cdot (0 {(0^{2} - 25)}^{2})$

Step 2.2.2

Remove parentheses.

$y = - \frac{1}{3} \cdot (0 {(0^{2} - 25)}^{2})$

Step 2.2.3

Remove parentheses.

$y = - \frac{1}{3} \cdot ((0) {({(0)}^{2} - 25)}^{2})$

Step 2.2.4

Simplify $- \frac{1}{3} \cdot ((0) {({(0)}^{2} - 25)}^{2})$ .

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Step 2.2.4.1

Cancel the common factor of $3$ .

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Step 2.2.4.1.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y = \frac{- 1}{3} \cdot ((0) {({(0)}^{2} - 25)}^{2})$

Step 2.2.4.1.2

Factor $3$ out of $(0) {({(0)}^{2} - 25)}^{2}$ .

$y = \frac{- 1}{3} \cdot (3 ((0) {({(0)}^{2} - 25)}^{2}))$

Step 2.2.4.1.3

Cancel the common factor.

$y = \frac{- 1}{3} \cdot (3 ((0) {({(0)}^{2} - 25)}^{2}))$

Step 2.2.4.1.4

Rewrite the expression.

$y = - 1 \cdot ((0) {({(0)}^{2} - 25)}^{2})$

Step 2.2.4.2

Multiply by zero.

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Step 2.2.4.2.1

Multiply $0$ by ${({(0)}^{2} - 25)}^{2}$ .

$y = - 1 \cdot 0$

Step 2.2.4.2.2

Multiply $- 1$ by $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 5, 0), (5, 0)$

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

Step 4