Find the x and y Intercepts f(x)=2(x-1)^2(5x+1)^2(x+2)(x^2+16)(x+2)

$f (x) = 2 {(x - 1)}^{2} {(5 x + 1)}^{2} (x + 2) (x^{2} + 16) (x + 2)$

Step 1

Find the x-intercepts.

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

$0 = 2 {(x - 1)}^{2} {(5 x + 1)}^{2} (x + 2) (x^{2} + 16) (x + 2)$

Solve the equation.

Rewrite the equation as $2 {(x - 1)}^{2} {(5 x + 1)}^{2} (x + 2) (x^{2} + 16) (x + 2) = 0$ .

$2 {(x - 1)}^{2} {(5 x + 1)}^{2} (x + 2) (x^{2} + 16) (x + 2) = 0$

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

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

${(5 x + 1)}^{2} = 0$

$x + 2 = 0$

$x^{2} + 16 = 0$

$x + 2 = 0$

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{2}$ equal to $0$ .

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

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set ${(5 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(5 x + 1)}^{2}$ equal to $0$ .

${(5 x + 1)}^{2} = 0$

Solve ${(5 x + 1)}^{2} = 0$ for $x$ .

Set the $5 x + 1$ equal to $0$ .

$5 x + 1 = 0$

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Divide each term in $5 x = - 1$ by $5$ and simplify.

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x^{2} + 16$ equal to $0$ and solve for $x$ .

Set $x^{2} + 16$ equal to $0$ .

$x^{2} + 16 = 0$

Solve $x^{2} + 16 = 0$ for $x$ .

Subtract $16$ from both sides of the equation.

$x^{2} = - 16$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 16}$

Simplify $\pm \sqrt{- 16}$ .

Rewrite $- 16$ as $- 1 (16)$ .

$x = \pm \sqrt{- 1 (16)}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{16}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{16}$

Rewrite $16$ as $4^{2}$ .

$x = \pm i \cdot \sqrt{4^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 4$

Move $4$ to the left of $i$ .

$x = \pm 4 i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 4 i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4 i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4 i, - 4 i$

The final solution is all the values that make $2 {(x - 1)}^{2} {(5 x + 1)}^{2} (x + 2) (x^{2} + 16) (x + 2) = 0$ true.

$x = 1, - \frac{1}{5}, - 2, 4 i, - 4 i$

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (- \frac{1}{5}, 0), (- 2, 0)$

Step 2

Find the y-intercepts.

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

$y = 2 {((0) - 1)}^{2} {(5 (0) + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Solve the equation.

Remove parentheses.

$y = 2 {(0 - 1)}^{2} {(5 (0) + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Remove parentheses.

$y = 2 {(0 - 1)}^{2} {(5 (0) + 1)}^{2} (0 + 2) ({(0)}^{2} + 16) ((0) + 2)$

Remove parentheses.

$y = 2 {(0 - 1)}^{2} {(5 (0) + 1)}^{2} (0 + 2) (0^{2} + 16) ((0) + 2)$

Remove parentheses.

$y = 2 {(0 - 1)}^{2} {(5 (0) + 1)}^{2} (0 + 2) (0^{2} + 16) (0 + 2)$

Remove parentheses.

$y = 2 {((0) - 1)}^{2} {(5 (0) + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Simplify $2 {((0) - 1)}^{2} {(5 (0) + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$ .

Subtract $1$ from $0$ .

$y = 2 {(- 1)}^{2} {(5 (0) + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

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

$y = 2 \cdot 1 {(5 (0) + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Multiply $2$ by $1$ .

$y = 2 {(5 (0) + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Multiply $5$ by $0$ .

$y = 2 {(0 + 1)}^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Add $0$ and $1$ .

$y = 2 \cdot 1^{2} ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

One to any power is one.

$y = 2 \cdot 1 ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Multiply $2$ by $1$ .

$y = 2 ((0) + 2) ({(0)}^{2} + 16) ((0) + 2)$

Add $0$ and $2$ .

$y = 2 \cdot 2 ({(0)}^{2} + 16) ((0) + 2)$

Multiply $2$ by $2$ .

$y = 4 ({(0)}^{2} + 16) ((0) + 2)$

Raising $0$ to any positive power yields $0$ .

$y = 4 (0 + 16) ((0) + 2)$

Add $0$ and $16$ .

$y = 4 \cdot 16 ((0) + 2)$

Multiply $4$ by $16$ .

$y = 64 ((0) + 2)$

Add $0$ and $2$ .

$y = 64 \cdot 2$

Multiply $64$ by $2$ .

$y = 128$

y-intercept(s) in point form.

y-intercept(s): $(0, 128)$

Step 3

List the intersections.

x-intercept(s): $(1, 0), (- \frac{1}{5}, 0), (- 2, 0)$

y-intercept(s): $(0, 128)$

Step 4

Find the x and y Intercepts f(x)=2(x-1)^2(5x+1)^2(x+2)(x^2+16)(x+2)

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