Find the x and y Intercepts f(x)=(x+5)(4x+1)(x+5)(x^2+9)

$f (x) = (x + 5) (4 x + 1) (x + 5) (x^{2} + 9)$

Step 1

Find the x-intercepts.

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

$0 = (x + 5) (4 x + 1) (x + 5) (x^{2} + 9)$

Solve the equation.

Rewrite the equation as $(x + 5) (4 x + 1) (x + 5) (x^{2} + 9) = 0$ .

$(x + 5) (4 x + 1) (x + 5) (x^{2} + 9) = 0$

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

$x + 5 = 0$

$4 x + 1 = 0$

$x + 5 = 0$

$x^{2} + 9 = 0$

Set $x + 5$ equal to $0$ and solve for $x$ .

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set $4 x + 1$ equal to $0$ and solve for $x$ .

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Solve $4 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Solve $x^{2} + 9 = 0$ for $x$ .

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

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

$x = \pm \sqrt{- 9}$

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$x = \pm i \cdot \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

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

$x = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$x = \pm 3 i$

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.

$x = 3 i$

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

$x = - 3 i$

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

$x = 3 i, - 3 i$

The final solution is all the values that make $(x + 5) (4 x + 1) (x + 5) (x^{2} + 9) = 0$ true.

$x = - 5, - \frac{1}{4}, 3 i, - 3 i$

x-intercept(s) in point form.

x-intercept(s): $(- 5, 0), (- \frac{1}{4}, 0)$

Step 2

Find the y-intercepts.

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

$y = ((0) + 5) (4 (0) + 1) ((0) + 5) ({(0)}^{2} + 9)$

Solve the equation.

Remove parentheses.

$y = (0 + 5) (4 (0) + 1) ((0) + 5) ({(0)}^{2} + 9)$

Remove parentheses.

$y = (0 + 5) (4 (0) + 1) (0 + 5) ({(0)}^{2} + 9)$

Remove parentheses.

$y = (0 + 5) (4 (0) + 1) (0 + 5) (0^{2} + 9)$

Remove parentheses.

$y = ((0) + 5) (4 (0) + 1) ((0) + 5) ({(0)}^{2} + 9)$

Simplify $((0) + 5) (4 (0) + 1) ((0) + 5) ({(0)}^{2} + 9)$ .

Add $0$ and $5$ .

$y = 5 (4 (0) + 1) ((0) + 5) ({(0)}^{2} + 9)$

Multiply $4$ by $0$ .

$y = 5 (0 + 1) ((0) + 5) ({(0)}^{2} + 9)$

Add $0$ and $1$ .

$y = 5 \cdot 1 ((0) + 5) ({(0)}^{2} + 9)$

Multiply $5$ by $1$ .

$y = 5 ((0) + 5) ({(0)}^{2} + 9)$

Add $0$ and $5$ .

$y = 5 \cdot 5 ({(0)}^{2} + 9)$

Multiply $5$ by $5$ .

$y = 25 ({(0)}^{2} + 9)$

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

$y = 25 (0 + 9)$

Add $0$ and $9$ .

$y = 25 \cdot 9$

Multiply $25$ by $9$ .

$y = 225$

y-intercept(s) in point form.

y-intercept(s): $(0, 225)$

Step 3

List the intersections.

x-intercept(s): $(- 5, 0), (- \frac{1}{4}, 0)$

y-intercept(s): $(0, 225)$

Step 4

Find the x and y Intercepts f(x)=(x+5)(4x+1)(x+5)(x^2+9)

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