Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

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 $0$ .

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

$x + 2 = 0$

$x - 1 = 0$

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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Step 1.2.3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.4

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

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Step 1.2.4.1

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

$x + 2 = 0$

Step 1.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

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

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

$x = 3, - 2, 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 2, 0), (1, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

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

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Step 2.2.5.1

Subtract $3$ from $0$ .

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

Step 2.2.5.2

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

$y = - 1 \cdot 9 ((0) + 2) ((0) - 1)$

Step 2.2.5.3

Multiply $- 1$ by $9$ .

$y = - 9 ((0) + 2) ((0) - 1)$

Step 2.2.5.4

Add $0$ and $2$ .

$y = - 9 \cdot 2 ((0) - 1)$

Step 2.2.5.5

Multiply $- 9$ by $2$ .

$y = - 18 ((0) - 1)$

Step 2.2.5.6

Subtract $1$ from $0$ .

$y = - 18 \cdot - 1$

Step 2.2.5.7

Multiply $- 18$ by $- 1$ .

$y = 18$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 18)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (- 2, 0), (1, 0)$

y-intercept(s): $(0, 18)$

Step 4