Find the x and y Intercepts y=x^2(x+3)(x+1)(x-1)(x-3)

$y = x^{2} (x + 3) (x + 1) (x - 1) (x - 3)$

Step 1

Find the x-intercepts.

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

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

Solve the equation.

Rewrite the equation as $x^{2} (x + 3) (x + 1) (x - 1) (x - 3) = 0$ .

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

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

$x^{2} = 0$

$x + 3 = 0$

$x + 1 = 0$

$x - 1 = 0$

$x - 3 = 0$

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 square 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 $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $x^{2} (x + 3) (x + 1) (x - 1) (x - 3) = 0$ true.

$x = 0, - 3, - 1, 1, 3$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 3, 0), (- 1, 0), (1, 0), (3, 0)$

Step 2

Find the y-intercepts.

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

$y = {(0)}^{2} ((0) + 3) ((0) + 1) ((0) - 1) ((0) - 3)$

Solve the equation.

Remove parentheses.

$y = 0^{2} ((0) + 3) ((0) + 1) ((0) - 1) ((0) - 3)$

Remove parentheses.

$y = 0^{2} (0 + 3) ((0) + 1) ((0) - 1) ((0) - 3)$

Remove parentheses.

$y = 0^{2} (0 + 3) (0 + 1) ((0) - 1) ((0) - 3)$

Remove parentheses.

$y = 0^{2} (0 + 3) (0 + 1) (0 - 1) ((0) - 3)$

Remove parentheses.

$y = 0^{2} (0 + 3) (0 + 1) (0 - 1) (0 - 3)$

Remove parentheses.

$y = {(0)}^{2} ((0) + 3) ((0) + 1) ((0) - 1) ((0) - 3)$

Simplify ${(0)}^{2} ((0) + 3) ((0) + 1) ((0) - 1) ((0) - 3)$ .

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

$y = 0 ((0) + 3) ((0) + 1) ((0) - 1) ((0) - 3)$

Add $0$ and $3$ .

$y = 0 \cdot 3 ((0) + 1) ((0) - 1) ((0) - 3)$

Multiply $0$ by $3$ .

$y = 0 ((0) + 1) ((0) - 1) ((0) - 3)$

Add $0$ and $1$ .

$y = 0 \cdot 1 ((0) - 1) ((0) - 3)$

Multiply $0$ by $1$ .

$y = 0 ((0) - 1) ((0) - 3)$

Subtract $1$ from $0$ .

$y = 0 \cdot - 1 ((0) - 3)$

Multiply $0$ by $- 1$ .

$y = 0 ((0) - 3)$

Subtract $3$ from $0$ .

$y = 0 \cdot - 3$

Multiply $0$ by $- 3$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 3, 0), (- 1, 0), (1, 0), (3, 0)$

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

Step 4

Find the x and y Intercepts y=x^2(x+3)(x+1)(x-1)(x-3)

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