Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

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 $0$ .

$x^{2} = 0$

$x + 3 = 0$

$x + 1 = 0$

$x - 1 = 0$

$x - 3 = 0$

Step 1.2.3

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

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Step 1.2.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 1.2.3.2

Solve $x^{2} = 0$ for $x$ .

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Step 1.2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 1.2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

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

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

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

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Step 1.2.6.1

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

$x - 1 = 0$

Step 1.2.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

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

$x - 3 = 0$

Step 1.2.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.8

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

$x = 0, - 3, - 1, 1, 3$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

Remove parentheses.

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

Step 2.2.6

Remove parentheses.

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

Step 2.2.7

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

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Step 2.2.7.1

Raising $0$ to any positive power yields $0$ .

$y = 0 ((0) + 3) ((0) + 1) ((0) - 1) ((0) - 3)$

Step 2.2.7.2

Add $0$ and $3$ .

$y = 0 \cdot 3 ((0) + 1) ((0) - 1) ((0) - 3)$

Step 2.2.7.3

Multiply $0$ by $3$ .

$y = 0 ((0) + 1) ((0) - 1) ((0) - 3)$

Step 2.2.7.4

Add $0$ and $1$ .

$y = 0 \cdot 1 ((0) - 1) ((0) - 3)$

Step 2.2.7.5

Multiply $0$ by $1$ .

$y = 0 ((0) - 1) ((0) - 3)$

Step 2.2.7.6

Subtract $1$ from $0$ .

$y = 0 \cdot - 1 ((0) - 3)$

Step 2.2.7.7

Multiply $0$ by $- 1$ .

$y = 0 ((0) - 3)$

Step 2.2.7.8

Subtract $3$ from $0$ .

$y = 0 \cdot - 3$

Step 2.2.7.9

Multiply $0$ by $- 3$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4