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

$f (x) = - 2 x (x - 1) (x - 1) {(x - 2)}^{3}$

Find the x-intercepts.

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

$0 = - 2 x (x - 1) (x - 1) {(x - 2)}^{3}$

Solve the equation.

Rewrite the equation as $- 2 x (x - 1) (x - 1) {(x - 2)}^{3} = 0$ .

$- 2 x (x - 1) (x - 1) {(x - 2)}^{3} = 0$

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

$x = 0$

$x - 1 = 0$

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

Set $x$ equal to $0$ .

$x = 0$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set ${(x - 2)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{3}$ equal to $0$ .

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

Solve ${(x - 2)}^{3} = 0$ for $x$ .

Set the $x - 2$ 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 $- 2 x (x - 1) (x - 1) {(x - 2)}^{3} = 0$ true.

$x = 0, 1, 2$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (1, 0), (2, 0)$

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

Simplify $- 2 \cdot (0 ((0) - 1) ((0) - 1) {((0) - 2)}^{3})$ .

Subtract $1$ from $0$ .

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

Multiply $0$ by $- 1$ .

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

Subtract $1$ from $0$ .

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

Multiply $0$ by $- 1$ .

$y = - 2 \cdot (0 {((0) - 2)}^{3})$

Subtract $2$ from $0$ .

$y = - 2 \cdot (0 {(- 2)}^{3})$

Multiply $- 2$ by ${(- 2)}^{3}$ by adding the exponents.

Move ${(- 2)}^{3}$ .

$y = {(- 2)}^{3} \cdot - 2 \cdot 0$

Multiply ${(- 2)}^{3}$ by $- 2$ .

Raise $- 2$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $3$ and $1$ .

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

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

$y = 16 \cdot 0$

Multiply $16$ by $0$ .

$y = 0$

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $(0, 0), (1, 0), (2, 0)$

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

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

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