Find the x and y Intercepts f(x)=(x^2+25)(6x+1)^2

$f (x) = (x^{2} + 25) {(6 x + 1)}^{2}$

Step 1

Find the x-intercepts.

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

$0 = (x^{2} + 25) {(6 x + 1)}^{2}$

Solve the equation.

Rewrite the equation as $(x^{2} + 25) {(6 x + 1)}^{2} = 0$ .

$(x^{2} + 25) {(6 x + 1)}^{2} = 0$

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

$x^{2} + 25 = 0$

${(6 x + 1)}^{2} = 0$

Set $x^{2} + 25$ equal to $0$ and solve for $x$ .

Set $x^{2} + 25$ equal to $0$ .

$x^{2} + 25 = 0$

Solve $x^{2} + 25 = 0$ for $x$ .

Subtract $25$ from both sides of the equation.

$x^{2} = - 25$

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

$x = \pm \sqrt{- 25}$

Simplify $\pm \sqrt{- 25}$ .

Rewrite $- 25$ as $- 1 (25)$ .

$x = \pm \sqrt{- 1 (25)}$

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

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

$x = \pm i \cdot \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

$x = \pm i \cdot \sqrt{5^{2}}$

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

$x = \pm i \cdot 5$

Move $5$ to the left of $i$ .

$x = \pm 5 i$

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.

$x = 5 i$

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

$x = - 5 i$

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

$x = 5 i, - 5 i$

Set ${(6 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(6 x + 1)}^{2}$ equal to $0$ .

${(6 x + 1)}^{2} = 0$

Solve ${(6 x + 1)}^{2} = 0$ for $x$ .

Set the $6 x + 1$ equal to $0$ .

$6 x + 1 = 0$

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Divide each term in $6 x = - 1$ by $6$ and simplify.

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

The final solution is all the values that make $(x^{2} + 25) {(6 x + 1)}^{2} = 0$ true.

$x = 5 i, - 5 i, - \frac{1}{6}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{6}, 0)$

Step 2

Find the y-intercepts.

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

$y = ({(0)}^{2} + 25) {(6 (0) + 1)}^{2}$

Solve the equation.

Remove parentheses.

$y = (0^{2} + 25) {(6 (0) + 1)}^{2}$

Remove parentheses.

$y = ({(0)}^{2} + 25) {(6 (0) + 1)}^{2}$

Simplify $({(0)}^{2} + 25) {(6 (0) + 1)}^{2}$ .

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

$y = (0 + 25) {(6 (0) + 1)}^{2}$

Add $0$ and $25$ .

$y = 25 {(6 (0) + 1)}^{2}$

Multiply $6$ by $0$ .

$y = 25 {(0 + 1)}^{2}$

Add $0$ and $1$ .

$y = 25 \cdot 1^{2}$

One to any power is one.

$y = 25 \cdot 1$

Multiply $25$ by $1$ .

$y = 25$

y-intercept(s) in point form.

y-intercept(s): $(0, 25)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{6}, 0)$

y-intercept(s): $(0, 25)$

Step 4

Find the x and y Intercepts f(x)=(x^2+25)(6x+1)^2

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