Find the x and y Intercepts f(x)=(2x^3+10x^2+8x)/(x^3+7x^2+12x)

$f (x) = \frac{2 x^{3} + 10 x^{2} + 8 x}{x^{3} + 7 x^{2} + 12 x}$

Step 1

Find the x-intercepts.

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

$0 = \frac{2 x^{3} + 10 x^{2} + 8 x}{x^{3} + 7 x^{2} + 12 x}$

Solve the equation.

Set the numerator equal to zero.

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

Solve the equation for $x$ .

Factor the left side of the equation.

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

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

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

Factor $2 x$ out of $10 x^{2}$ .

$2 x (x^{2}) + 2 x (5 x) + 8 x = 0$

Factor $2 x$ out of $8 x$ .

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

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

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

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

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

Factor.

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

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 $4$ and whose sum is $5$ .

$1, 4$

Write the factored form using these integers.

$2 x ((x + 1) (x + 4)) = 0$

Remove unnecessary parentheses.

$2 x (x + 1) (x + 4) = 0$

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

$x = 0$

$x + 1 = 0$

$x + 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $2 x (x + 1) (x + 4) = 0$ true.

$x = 0, - 1, - 4$

Exclude the solutions that do not make $0 = \frac{2 x^{3} + 10 x^{2} + 8 x}{x^{3} + 7 x^{2} + 12 x}$ true.

$x = - 1$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = \frac{2 {(0)}^{3} + 10 {(0)}^{2} + 8 (0)}{{(0)}^{3} + 7 {(0)}^{2} + 12 (0)}$

The equation has an undefined fraction.

Undefined

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): $(- 1, 0)$

y-intercept(s): $None$

Step 4

Find the x and y Intercepts f(x)=(2x^3+10x^2+8x)/(x^3+7x^2+12x)

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