Algebra Examples

$f (x) = x^{2} (x + 1) (x^{2} - 5) (x - 9) (3 x + 5) (x + 1)$

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 $x$ .

$0 = x^{2} (x + 1) (x^{2} - 5) (x - 9) (3 x + 5) (x + 1)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{2} (x + 1) (x^{2} - 5) (x - 9) (3 x + 5) (x + 1) = 0$ .

$x^{2} (x + 1) (x^{2} - 5) (x - 9) (3 x + 5) (x + 1) = 0$

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 $0$ .

$x^{2} = 0$

$x + 1 = 0$

$x^{2} - 5 = 0$

$x - 9 = 0$

$3 x + 5 = 0$

$x + 1 = 0$

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 $x + 1$ 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 $x^{2} - 5$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Step 1.2.5.2

Solve $x^{2} - 5 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $5$ to both sides of the equation.

$x^{2} = 5$

Step 1.2.5.2.2

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

$x = \pm \sqrt{5}$

Step 1.2.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{5}$

Step 1.2.5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{5}$

Step 1.2.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{5}, - \sqrt{5}$

Step 1.2.6

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 1.2.6.2

Add $9$ to both sides of the equation.

$x = 9$

Step 1.2.7

Set $3 x + 5$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 1.2.7.2

Solve $3 x + 5 = 0$ for $x$ .

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Step 1.2.7.2.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 1.2.7.2.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 1.2.7.2.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 1.2.7.2.2.2

Simplify the left side.

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Step 1.2.7.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 1.2.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 1.2.7.2.2.3

Simplify the right side.

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Step 1.2.7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 1.2.8

The final solution is all the values that make $x^{2} (x + 1) (x^{2} - 5) (x - 9) (3 x + 5) (x + 1) = 0$ true.

$x = 0, - 1, \sqrt{5}, - \sqrt{5}, 9, - \frac{5}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 1, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0), (9, 0), (- \frac{5}{3}, 0)$

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 $y$ .

$y = {(0)}^{2} ((0) + 1) ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{2} ((0) + 1) ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.2

Remove parentheses.

$y = 0^{2} (0 + 1) ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.3

Remove parentheses.

$y = 0^{2} (0 + 1) (0^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.4

Remove parentheses.

$y = 0^{2} (0 + 1) (0^{2} - 5) (0 - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.5

Remove parentheses.

$y = 0^{2} (0 + 1) (0^{2} - 5) (0 - 9) (3 (0) + 5) (0 + 1)$

Step 2.2.6

Remove parentheses.

$y = {(0)}^{2} ((0) + 1) ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.7

Simplify ${(0)}^{2} ((0) + 1) ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$ .

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Step 2.2.7.1

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

$y = 0 ((0) + 1) ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.7.2

Add $0$ and $1$ .

$y = 0 \cdot 1 ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.7.3

Multiply $0$ by $1$ .

$y = 0 ({(0)}^{2} - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.7.4

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

$y = 0 (0 - 5) ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.7.5

Subtract $5$ from $0$ .

$y = 0 \cdot - 5 ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.7.6

Multiply $0$ by $- 5$ .

$y = 0 ((0) - 9) (3 (0) + 5) ((0) + 1)$

Step 2.2.7.7

Subtract $9$ from $0$ .

$y = 0 \cdot - 9 (3 (0) + 5) ((0) + 1)$

Step 2.2.7.8

Multiply $0$ by $- 9$ .

$y = 0 (3 (0) + 5) ((0) + 1)$

Step 2.2.7.9

Multiply $3$ by $0$ .

$y = 0 (0 + 5) ((0) + 1)$

Step 2.2.7.10

Add $0$ and $5$ .

$y = 0 \cdot 5 ((0) + 1)$

Step 2.2.7.11

Multiply $0$ by $5$ .

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

Step 2.2.7.12

Add $0$ and $1$ .

$y = 0 \cdot 1$

Step 2.2.7.13

Multiply $0$ by $1$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 1, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0), (9, 0), (- \frac{5}{3}, 0)$

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

Step 4