Find the x and y Intercepts f(x)=-5(x+1)(x^2-3)(x+1)(4x+5)^2

$f (x) = - 5 (x + 1) (x^{2} - 3) (x + 1) {(4 x + 5)}^{2}$

Step 1

Find the x-intercepts.

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

$0 = - 5 (x + 1) (x^{2} - 3) (x + 1) {(4 x + 5)}^{2}$

Solve the equation.

Rewrite the equation as $- 5 (x + 1) (x^{2} - 3) (x + 1) {(4 x + 5)}^{2} = 0$ .

$- 5 (x + 1) (x^{2} - 3) (x + 1) {(4 x + 5)}^{2} = 0$

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

$x + 1 = 0$

$x^{2} - 3 = 0$

$x + 1 = 0$

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

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

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

$x + 1 = 0$

Subtract $1$ from 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 $x$ .

Add $3$ to both sides of the equation.

$x^{2} = 3$

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

$x = \pm \sqrt{3}$

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.

$x = \sqrt{3}$

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

$x = - \sqrt{3}$

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

$x = \sqrt{3}, - \sqrt{3}$

Set ${(4 x + 5)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(4 x + 5)}^{2}$ equal to $0$ .

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

Solve ${(4 x + 5)}^{2} = 0$ for $x$ .

Set the $4 x + 5$ equal to $0$ .

$4 x + 5 = 0$

Solve for $x$ .

Subtract $5$ from both sides of the equation.

$4 x = - 5$

Divide each term in $4 x = - 5$ by $4$ and simplify.

Divide each term in $4 x = - 5$ by $4$ .

$\frac{4 x}{4} = \frac{- 5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 5}{4}$

Divide $x$ by $1$ .

$x = \frac{- 5}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{4}$

The final solution is all the values that make $- 5 (x + 1) (x^{2} - 3) (x + 1) {(4 x + 5)}^{2} = 0$ true.

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

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (\sqrt{3}, 0), (- \sqrt{3}, 0), (- \frac{5}{4}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 5 ((0) + 1) ({(0)}^{2} - 3) ((0) + 1) {(4 (0) + 5)}^{2}$

Solve the equation.

Remove parentheses.

$y = - 5 (0 + 1) ({(0)}^{2} - 3) ((0) + 1) {(4 (0) + 5)}^{2}$

Remove parentheses.

$y = - 5 (0 + 1) (0^{2} - 3) ((0) + 1) {(4 (0) + 5)}^{2}$

Remove parentheses.

$y = - 5 (0 + 1) (0^{2} - 3) (0 + 1) {(4 (0) + 5)}^{2}$

Remove parentheses.

$y = - 5 ((0) + 1) ({(0)}^{2} - 3) ((0) + 1) {(4 (0) + 5)}^{2}$

Simplify $- 5 ((0) + 1) ({(0)}^{2} - 3) ((0) + 1) {(4 (0) + 5)}^{2}$ .

Add $0$ and $1$ .

$y = - 5 \cdot 1 ({(0)}^{2} - 3) ((0) + 1) {(4 (0) + 5)}^{2}$

Multiply $- 5$ by $1$ .

$y = - 5 ({(0)}^{2} - 3) ((0) + 1) {(4 (0) + 5)}^{2}$

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

$y = - 5 (0 - 3) ((0) + 1) {(4 (0) + 5)}^{2}$

Subtract $3$ from $0$ .

$y = - 5 \cdot - 3 ((0) + 1) {(4 (0) + 5)}^{2}$

Multiply $- 5$ by $- 3$ .

$y = 15 ((0) + 1) {(4 (0) + 5)}^{2}$

Add $0$ and $1$ .

$y = 15 \cdot 1 {(4 (0) + 5)}^{2}$

Multiply $15$ by $1$ .

$y = 15 {(4 (0) + 5)}^{2}$

Multiply $4$ by $0$ .

$y = 15 {(0 + 5)}^{2}$

Add $0$ and $5$ .

$y = 15 \cdot 5^{2}$

Raise $5$ to the power of $2$ .

$y = 15 \cdot 25$

Multiply $15$ by $25$ .

$y = 375$

y-intercept(s) in point form.

y-intercept(s): $(0, 375)$

Step 3

List the intersections.

x-intercept(s): $(- 1, 0), (\sqrt{3}, 0), (- \sqrt{3}, 0), (- \frac{5}{4}, 0)$

y-intercept(s): $(0, 375)$

Step 4

Find the x and y Intercepts f(x)=-5(x+1)(x^2-3)(x+1)(4x+5)^2

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