Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

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 $0$ .

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

$x + 1 = 0$

$x - 1 = 0$

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

$x + 2 = 0$

Step 1.2.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

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

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.5

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

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Step 1.2.5.1

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

$x - 1 = 0$

Step 1.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.6

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

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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Step 1.2.6.2.1

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

$x - 2 = 0$

Step 1.2.6.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.7

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

$x = - 2, - 1, 1, 2$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

Remove parentheses.

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

Step 2.2.6

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

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Step 2.2.6.1

Add $0$ and $2$ .

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

Step 2.2.6.2

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

$y = 4 ((0) + 1) ((0) - 1) {((0) - 2)}^{2}$

Step 2.2.6.3

Add $0$ and $1$ .

$y = 4 \cdot 1 ((0) - 1) {((0) - 2)}^{2}$

Step 2.2.6.4

Multiply $4$ by $1$ .

$y = 4 ((0) - 1) {((0) - 2)}^{2}$

Step 2.2.6.5

Subtract $1$ from $0$ .

$y = 4 \cdot - 1 {((0) - 2)}^{2}$

Step 2.2.6.6

Multiply $4$ by $- 1$ .

$y = - 4 {((0) - 2)}^{2}$

Step 2.2.6.7

Subtract $2$ from $0$ .

$y = - 4 {(- 2)}^{2}$

Step 2.2.6.8

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

$y = - 4 \cdot 4$

Step 2.2.6.9

Multiply $- 4$ by $4$ .

$y = - 16$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 16)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 16)$

Step 4