Algebra Examples

$f (x) = \frac{2 x^{3} + 8 x^{2} - 10 x}{2 x^{2} + 2 x}$

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 $x$ .

$0 = \frac{2 x^{3} + 8 x^{2} - 10 x}{2 x^{2} + 2 x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$2 x^{3} + 8 x^{2} - 10 x = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor the left side of the equation.

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Step 1.2.2.1.1

Factor $2 x$ out of $2 x^{3} + 8 x^{2} - 10 x$ .

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Step 1.2.2.1.1.1

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

$2 x (x^{2}) + 8 x^{2} - 10 x = 0$

Step 1.2.2.1.1.2

Factor $2 x$ out of $8 x^{2}$ .

$2 x (x^{2}) + 2 x (4 x) - 10 x = 0$

Step 1.2.2.1.1.3

Factor $2 x$ out of $- 10 x$ .

$2 x (x^{2}) + 2 x (4 x) + 2 x (- 5) = 0$

Step 1.2.2.1.1.4

Factor $2 x$ out of $2 x (x^{2}) + 2 x (4 x)$ .

$2 x (x^{2} + 4 x) + 2 x (- 5) = 0$

Step 1.2.2.1.1.5

Factor $2 x$ out of $2 x (x^{2} + 4 x) + 2 x (- 5)$ .

$2 x (x^{2} + 4 x - 5) = 0$

Step 1.2.2.1.2

Factor.

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Step 1.2.2.1.2.1

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 1.2.2.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 1.2.2.1.2.1.2

Write the factored form using these integers.

$2 x ((x - 1) (x + 5)) = 0$

Step 1.2.2.1.2.2

Remove unnecessary parentheses.

$2 x (x - 1) (x + 5) = 0$

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 $0$ .

$x = 0$

$x - 1 = 0$

$x + 5 = 0$

Step 1.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.2.4

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

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Step 1.2.2.4.1

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

$x - 1 = 0$

Step 1.2.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.2.5

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

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Step 1.2.2.5.1

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

$x + 5 = 0$

Step 1.2.2.5.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.2.6

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

$x = 0, 1, - 5$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{2 x^{3} + 8 x^{2} - 10 x}{2 x^{2} + 2 x}$ true.

$x = 1, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (- 5, 0)$

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 $y$ .

$y = \frac{2 {(0)}^{3} + 8 {(0)}^{2} - 10 \cdot 0}{2 {(0)}^{2} + 2 (0)}$

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): $(1, 0), (- 5, 0)$

y-intercept(s): $None$

Step 4