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

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

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

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

Solve the equation for $x$ .

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

$x + 2 = 0$

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

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.

$x = - 2$

Set ${(x + 1)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x + 1)}^{3}$ equal to $0$ .

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

Solve ${(x + 1)}^{3} = 0$ for $x$ .

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$x = - 2, - 1$

Exclude the solutions that do not make $0 = \frac{(x + 2) {(x + 1)}^{3}}{(x + 2) {(x - 3)}^{2}}$ true.

$x = - 1$

x-intercept(s) in point form.

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

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

Cancel the common factor of $(0) + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Add $0$ and $1$ .

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

One to any power is one.

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

Simplify the denominator.

Subtract $3$ from $0$ .

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

Raise $- 3$ to the power of $2$ .

$y = \frac{1}{9}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{9})$

List the intersections.

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

y-intercept(s): $(0, \frac{1}{9})$

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

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