Algebra Examples

$P (x) = 5 {(x + 1)}^{2} {(x - 2)}^{3}$

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 $x$ .

$0 = 5 {(x + 1)}^{2} {(x - 2)}^{3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5 {(x + 1)}^{2} {(x - 2)}^{3} = 0$ .

$5 {(x + 1)}^{2} {(x - 2)}^{3} = 0$

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 $0$ .

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

${(x - 2)}^{3} = 0$

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

$x + 1 = 0$

Step 1.2.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.4

Set ${(x - 2)}^{3}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set ${(x - 2)}^{3}$ equal to $0$ .

${(x - 2)}^{3} = 0$

Step 1.2.4.2

Solve ${(x - 2)}^{3} = 0$ for $x$ .

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Step 1.2.4.2.1

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

$x - 2 = 0$

Step 1.2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5

The final solution is all the values that make $5 {(x + 1)}^{2} {(x - 2)}^{3} = 0$ true.

$x = - 1, 2$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = 5 {((0) + 1)}^{2} {((0) - 2)}^{3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 5 {(0 + 1)}^{2} {((0) - 2)}^{3}$

Step 2.2.2

Remove parentheses.

$y = 5 {(0 + 1)}^{2} {(0 - 2)}^{3}$

Step 2.2.3

Remove parentheses.

$y = 5 {((0) + 1)}^{2} {((0) - 2)}^{3}$

Step 2.2.4

Simplify $5 {((0) + 1)}^{2} {((0) - 2)}^{3}$ .

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Step 2.2.4.1

Add $0$ and $1$ .

$y = 5 \cdot 1^{2} {((0) - 2)}^{3}$

Step 2.2.4.2

One to any power is one.

$y = 5 \cdot 1 {((0) - 2)}^{3}$

Step 2.2.4.3

Multiply $5$ by $1$ .

$y = 5 {((0) - 2)}^{3}$

Step 2.2.4.4

Subtract $2$ from $0$ .

$y = 5 {(- 2)}^{3}$

Step 2.2.4.5

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

$y = 5 \cdot - 8$

Step 2.2.4.6

Multiply $5$ by $- 8$ .

$y = - 40$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 40)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 40)$

Step 4