Find the x and y Intercepts f(x)=((x+4)^2(x-2))/(2(x-1)^2(x-3))

$f (x) = \frac{{(x + 4)}^{2} (x - 2)}{2 {(x - 1)}^{2} (x - 3)}$

Step 1

Find the x-intercepts.

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

$0 = \frac{{(x + 4)}^{2} (x - 2)}{2 {(x - 1)}^{2} (x - 3)}$

Solve the equation.

Set the numerator equal to zero.

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

Solve the equation for $x$ .

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

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

$x - 2 = 0$

Set ${(x + 4)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x + 4)}^{2}$ equal to $0$ .

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

Solve ${(x + 4)}^{2} = 0$ for $x$ .

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$x = - 4, 2$

x-intercept(s) in point form.

x-intercept(s): $(- 4, 0), (2, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{{((0) + 4)}^{2} ((0) - 2)}{2 {((0) - 1)}^{2} ((0) - 3)}$

Solve the equation.

Remove parentheses.

$y = \frac{{(0 + 4)}^{2} ((0) - 2)}{2 {((0) - 1)}^{2} ((0) - 3)}$

Remove parentheses.

$y = \frac{{(0 + 4)}^{2} (0 - 2)}{2 {((0) - 1)}^{2} ((0) - 3)}$

Remove parentheses.

$y = \frac{{(0 + 4)}^{2} (0 - 2)}{2 {(0 - 1)}^{2} ((0) - 3)}$

Remove parentheses.

$y = \frac{{(0 + 4)}^{2} (0 - 2)}{2 {(0 - 1)}^{2} (0 - 3)}$

Remove parentheses.

$y = \frac{{((0) + 4)}^{2} ((0) - 2)}{2 {((0) - 1)}^{2} ((0) - 3)}$

Simplify $\frac{{((0) + 4)}^{2} ((0) - 2)}{2 {((0) - 1)}^{2} ((0) - 3)}$ .

Reduce the expression by cancelling the common factors.

Cancel the common factor of $(0) - 2$ and $2$ .

Factor $2$ out of ${((0) + 4)}^{2} ((0) - 2)$ .

$y = \frac{2 ({((0) + 4)}^{2} (0 - 1))}{2 {((0) - 1)}^{2} ((0) - 3)}$

Cancel the common factors.

Factor $2$ out of $2 {((0) - 1)}^{2} ((0) - 3)$ .

$y = \frac{2 ({((0) + 4)}^{2} (0 - 1))}{2 ({((0) - 1)}^{2} ((0) - 3))}$

Cancel the common factor.

$y = \frac{2 ({((0) + 4)}^{2} (0 - 1))}{2 ({((0) - 1)}^{2} ((0) - 3))}$

Rewrite the expression.

$y = \frac{{((0) + 4)}^{2} (0 - 1)}{{((0) - 1)}^{2} ((0) - 3)}$

Cancel the common factor of $0 - 1$ and ${((0) - 1)}^{2}$ .

Reorder terms.

$y = \frac{{((0) + 4)}^{2} (0 - 1)}{{(0 - 1)}^{2} ((0) - 3)}$

Factor $0 - 1$ out of ${((0) + 4)}^{2} (0 - 1)$ .

$y = \frac{(0 - 1) {((0) + 4)}^{2}}{{(0 - 1)}^{2} ((0) - 3)}$

Cancel the common factors.

Factor $0 - 1$ out of ${(0 - 1)}^{2} ((0) - 3)$ .

$y = \frac{(0 - 1) {((0) + 4)}^{2}}{(0 - 1) ((0 - 1) ((0) - 3))}$

Cancel the common factor.

$y = \frac{(0 - 1) {((0) + 4)}^{2}}{(0 - 1) ((0 - 1) ((0) - 3))}$

Rewrite the expression.

$y = \frac{{((0) + 4)}^{2}}{(0 - 1) ((0) - 3)}$

Simplify the numerator.

Add $0$ and $4$ .

$y = \frac{4^{2}}{(0 - 1) (0 - 3)}$

Raise $4$ to the power of $2$ .

$y = \frac{16}{(0 - 1) (0 - 3)}$

Simplify the denominator.

Subtract $1$ from $0$ .

$y = \frac{16}{- 1 (0 - 3)}$

Subtract $3$ from $0$ .

$y = \frac{16}{- 1 \cdot - 3}$

Multiply $- 1$ by $- 3$ .

$y = \frac{16}{3}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{16}{3})$

Step 3

List the intersections.

x-intercept(s): $(- 4, 0), (2, 0)$

y-intercept(s): $(0, \frac{16}{3})$

Step 4

Find the x and y Intercepts f(x)=((x+4)^2(x-2))/(2(x-1)^2(x-3))

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