Find the x and y Intercepts f(x)=x^2(x-5)^3(x+2)

$f (x) = x^{2} {(x - 5)}^{3} (x + 2)$

Step 1

Find the x-intercepts.

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

$0 = x^{2} {(x - 5)}^{3} (x + 2)$

Solve the equation.

Rewrite the equation as $x^{2} {(x - 5)}^{3} (x + 2) = 0$ .

$x^{2} {(x - 5)}^{3} (x + 2) = 0$

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

$x^{2} = 0$

${(x - 5)}^{3} = 0$

$x + 2 = 0$

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set ${(x - 5)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{3}$ equal to $0$ .

${(x - 5)}^{3} = 0$

Solve ${(x - 5)}^{3} = 0$ for $x$ .

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $x^{2} {(x - 5)}^{3} (x + 2) = 0$ true.

$x = 0, 5, - 2$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (5, 0), (- 2, 0)$

Step 2

Find the y-intercepts.

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

$y = {(0)}^{2} {((0) - 5)}^{3} ((0) + 2)$

Solve the equation.

Remove parentheses.

$y = 0^{2} {((0) - 5)}^{3} ((0) + 2)$

Remove parentheses.

$y = 0^{2} {(0 - 5)}^{3} ((0) + 2)$

Remove parentheses.

$y = 0^{2} {(0 - 5)}^{3} (0 + 2)$

Remove parentheses.

$y = {(0)}^{2} {((0) - 5)}^{3} ((0) + 2)$

Simplify ${(0)}^{2} {((0) - 5)}^{3} ((0) + 2)$ .

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

$y = 0 {((0) - 5)}^{3} ((0) + 2)$

Subtract $5$ from $0$ .

$y = 0 {(- 5)}^{3} ((0) + 2)$

Raise $- 5$ to the power of $3$ .

$y = 0 \cdot - 125 ((0) + 2)$

Multiply $0$ by $- 125$ .

$y = 0 ((0) + 2)$

Add $0$ and $2$ .

$y = 0 \cdot 2$

Multiply $0$ by $2$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (5, 0), (- 2, 0)$

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

Step 4

Find the x and y Intercepts f(x)=x^2(x-5)^3(x+2)

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