Find the x and y Intercepts f(x)=2x(x-2)^2(4x-2)^2

$f (x) = 2 x {(x - 2)}^{2} {(4 x - 2)}^{2}$

Step 1

Find the x-intercepts.

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

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

Solve the equation.

Rewrite the equation as $2 x {(x - 2)}^{2} {(4 x - 2)}^{2} = 0$ .

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

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

$x = 0$

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

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

Set $x$ equal to $0$ .

$x = 0$

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

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

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

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

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

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

Factor the left side of the equation.

Factor $2$ out of $4 x - 2$ .

Factor $2$ out of $4 x$ .

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

Factor $2$ out of $- 2$ .

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

Factor $2$ out of $2 (2 x) + 2 (- 1)$ .

${(2 (2 x - 1))}^{2} = 0$

Apply the product rule to $2 (2 x - 1)$ .

$2^{2} {(2 x - 1)}^{2} = 0$

Divide each term in $2^{2} {(2 x - 1)}^{2} = 0$ by $2^{2}$ and simplify.

Divide each term in $2^{2} {(2 x - 1)}^{2} = 0$ by $2^{2}$ .

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

Simplify the left side.

Cancel the common factor of $2^{2}$ .

Cancel the common factor.

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

Divide ${(2 x - 1)}^{2}$ by $1$ .

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

Simplify the right side.

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

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

Divide $0$ by $4$ .

${(2 x - 1)}^{2} = 0$

Set the $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

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

$x = 0, 2, \frac{1}{2}$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (2, 0), (\frac{1}{2}, 0)$

Step 2

Find the y-intercepts.

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

$y = 2 (0) {((0) - 2)}^{2} {(4 (0) - 2)}^{2}$

Solve the equation.

Remove parentheses.

$y = 2 (0) {(0 - 2)}^{2} {(4 (0) - 2)}^{2}$

Remove parentheses.

$y = 2 (0) {((0) - 2)}^{2} {(4 (0) - 2)}^{2}$

Simplify $2 (0) {((0) - 2)}^{2} {(4 (0) - 2)}^{2}$ .

Multiply $2$ by $0$ .

$y = 0 {((0) - 2)}^{2} {(4 (0) - 2)}^{2}$

Subtract $2$ from $0$ .

$y = 0 {(- 2)}^{2} {(4 (0) - 2)}^{2}$

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

$y = 0 \cdot 4 {(4 (0) - 2)}^{2}$

Multiply $0$ by $4$ .

$y = 0 {(4 (0) - 2)}^{2}$

Multiply $4$ by $0$ .

$y = 0 {(0 - 2)}^{2}$

Subtract $2$ from $0$ .

$y = 0 {(- 2)}^{2}$

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

$y = 0 \cdot 4$

Multiply $0$ by $4$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (2, 0), (\frac{1}{2}, 0)$

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

Step 4

Find the x and y Intercepts f(x)=2x(x-2)^2(4x-2)^2

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