Algebra Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

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

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

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

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

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

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

$x + 2 = 0$

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

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

$x + 2 = 0$

Step 1.2.2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.2.3

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

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Step 1.2.2.3.1

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

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

Step 1.2.2.3.2

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

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Step 1.2.2.3.2.1

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

$x + 1 = 0$

Step 1.2.2.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.2.4

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

$x = - 2, - 1$

Step 1.2.3

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

$x = - 1$

Step 1.3

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

Remove parentheses.

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

Step 2.2.6

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

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Step 2.2.6.1

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

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Step 2.2.6.1.1

Cancel the common factor.

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

Step 2.2.6.1.2

Rewrite the expression.

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

Step 2.2.6.2

Simplify the numerator.

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Step 2.2.6.2.1

Add $0$ and $1$ .

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

Step 2.2.6.2.2

One to any power is one.

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

Step 2.2.6.3

Simplify the denominator.

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Step 2.2.6.3.1

Subtract $3$ from $0$ .

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

Step 2.2.6.3.2

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

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

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4