Algebra Examples

$f (x) = \frac{3 x^{3} + 2 x^{2}}{x^{2} + x}$

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 $x$ .

$0 = \frac{3 x^{3} + 2 x^{2}}{x^{2} + x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$3 x^{3} + 2 x^{2} = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor $x^{2}$ out of $3 x^{3} + 2 x^{2}$ .

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Step 1.2.2.1.1

Factor $x^{2}$ out of $3 x^{3}$ .

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

Step 1.2.2.1.2

Factor $x^{2}$ out of $2 x^{2}$ .

$x^{2} (3 x) + x^{2} \cdot 2 = 0$

Step 1.2.2.1.3

Factor $x^{2}$ out of $x^{2} (3 x) + x^{2} \cdot 2$ .

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

Step 1.2.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$3 x + 2 = 0$

Step 1.2.2.3

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

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Step 1.2.2.3.1

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

$x^{2} = 0$

Step 1.2.2.3.2

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

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Step 1.2.2.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.2.3.2.2.1

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

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

Step 1.2.2.3.2.2.2

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

$x = \pm 0$

Step 1.2.2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.2.4

Set $3 x + 2$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Step 1.2.2.4.2

Solve $3 x + 2 = 0$ for $x$ .

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Step 1.2.2.4.2.1

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Step 1.2.2.4.2.2

Divide each term in $3 x = - 2$ by $3$ and simplify.

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Step 1.2.2.4.2.2.1

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Step 1.2.2.4.2.2.2

Simplify the left side.

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Step 1.2.2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Step 1.2.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Step 1.2.2.4.2.2.3

Simplify the right side.

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Step 1.2.2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Step 1.2.2.5

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

$x = 0, - \frac{2}{3}$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{3 x^{3} + 2 x^{2}}{x^{2} + x}$ true.

$x = - \frac{2}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{2}{3}, 0)$

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 $y$ .

$y = \frac{3 {(0)}^{3} + 2 {(0)}^{2}}{{(0)}^{2} + 0}$

Step 2.2

The equation has an undefined fraction.

Undefined

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(- \frac{2}{3}, 0)$

y-intercept(s): $None$

Step 4