Find the x and y Intercepts f(x)=-1/4x(x^2-25)^2

$f (x) = - \frac{1}{4} x {(x^{2} - 25)}^{2}$

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{1}{4} \cdot (x {(x^{2} - 25)}^{2})$

Solve the equation.

Rewrite the equation as $- \frac{1}{4} \cdot (x {(x^{2} - 25)}^{2}) = 0$ .

$- \frac{1}{4} \cdot (x {(x^{2} - 25)}^{2}) = 0$

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{1}{4} \cdot (x {(x^{2} - 25)}^{2})) = - 4 \cdot 0$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 4 (- \frac{1}{4} \cdot (x {(x^{2} - 25)}^{2}))$ .

Multiply $- \frac{1}{4} (x {(x^{2} - 25)}^{2})$ .

Combine $x$ and $\frac{1}{4}$ .

$- 4 (- \frac{x}{4} {(x^{2} - 25)}^{2}) = - 4 \cdot 0$

Combine ${(x^{2} - 25)}^{2}$ and $\frac{x}{4}$ .

$- 4 (- \frac{{(x^{2} - 25)}^{2} x}{4}) = - 4 \cdot 0$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{{(x^{2} - 25)}^{2} x}{4}$ into the numerator.

$- 4 \frac{- {(x^{2} - 25)}^{2} x}{4} = - 4 \cdot 0$

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- {(x^{2} - 25)}^{2} x}{4} = - 4 \cdot 0$

Cancel the common factor.

$4 \cdot - 1 \frac{- {(x^{2} - 25)}^{2} x}{4} = - 4 \cdot 0$

Rewrite the expression.

$- (- {(x^{2} - 25)}^{2} x) = - 4 \cdot 0$

Simplify the expression.

Multiply $- 1$ by $- 1$ .

$1 ({(x^{2} - 25)}^{2} x) = - 4 \cdot 0$

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

${(x^{2} - 25)}^{2} x = - 4 \cdot 0$

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

$x {(x^{2} - 25)}^{2} = - 4 \cdot 0$

Simplify the right side.

Multiply $- 4$ by $0$ .

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

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

$x = 0$

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

Set $x$ equal to $0$ .

$x = 0$

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

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

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

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

Factor the left side of the equation.

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

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

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 $b = 5$ .

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

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

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

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

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

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

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

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

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

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

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

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

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

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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

$x = - 5, 5$

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

$x = 0, - 5, 5$

x-intercept(s) in point form.

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

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{1}{4} \cdot ((0) {({(0)}^{2} - 25)}^{2})$

Solve the equation.

Remove parentheses.

$y = - \frac{1}{4} \cdot (0 {(0^{2} - 25)}^{2})$

Remove parentheses.

$y = - \frac{1}{4} \cdot (0 {(0^{2} - 25)}^{2})$

Remove parentheses.

$y = - \frac{1}{4} \cdot ((0) {({(0)}^{2} - 25)}^{2})$

Simplify $- \frac{1}{4} \cdot ((0) {({(0)}^{2} - 25)}^{2})$ .

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$y = \frac{- 1}{4} \cdot ((0) {({(0)}^{2} - 25)}^{2})$

Factor $4$ out of $(0) {({(0)}^{2} - 25)}^{2}$ .

$y = \frac{- 1}{4} \cdot (4 ((0) {({(0)}^{2} - 25)}^{2}))$

Cancel the common factor.

$y = \frac{- 1}{4} \cdot (4 ((0) {({(0)}^{2} - 25)}^{2}))$

Rewrite the expression.

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

Multiply by zero.

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

$y = - 1 \cdot 0$

Multiply $- 1$ by $0$ .

$y = 0$

y-intercept(s) in point form.

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

List the intersections.

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

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

Find the x and y Intercepts f(x)=-1/4x(x^2-25)^2

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