Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

$x = 0$

$x - 1 = 0$

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

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 1 = 0$

Step 1.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

$x + 2 = 0$

Step 1.2.5.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.6

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

$x = 0, 1, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (1, 0), (- 2, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

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

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Step 2.2.4.1

Subtract $1$ from $0$ .

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

Step 2.2.4.2

Multiply $0$ by $- 1$ .

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

Step 2.2.4.3

Add $0$ and $2$ .

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

Step 2.2.4.4

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

$y = 0 \cdot 4$

Step 2.2.4.5

Multiply $0$ by $4$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (1, 0), (- 2, 0)$

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

Step 4