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

$f (x) = x^{2} (x^{2} + 1) {(x + 2)}^{2}$

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

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

$x^{2} = 0$

$x^{2} + 1 = 0$

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

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 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 $x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Solve $x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

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

$x = \pm \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

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

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

$x = i$

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

$x = - i$

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

$x = i, - i$

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

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

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

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

Set the $x + 2$ 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 $x^{2} (x^{2} + 1) {(x + 2)}^{2} = 0$ true.

$x = 0, i, - i, - 2$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

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

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

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

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

Add $0$ and $1$ .

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

Multiply $0$ by $1$ .

$y = 0 {((0) + 2)}^{2}$

Add $0$ and $2$ .

$y = 0 \cdot 2^{2}$

Raise $2$ to the power of $2$ .

$y = 0 \cdot 4$

Multiply $0$ by $4$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4

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

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