Find the x and y Intercepts f(x)=x^3+8x^2+15x

$f (x) = x^{3} + 8 x^{2} + 15 x$

Step 1

Find the x-intercepts.

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

$0 = x^{3} + 8 x^{2} + 15 x$

Solve the equation.

Rewrite the equation as $x^{3} + 8 x^{2} + 15 x = 0$ .

$x^{3} + 8 x^{2} + 15 x = 0$

Factor the left side of the equation.

Factor $x$ out of $x^{3} + 8 x^{2} + 15 x$ .

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} + 8 x^{2} + 15 x = 0$

Factor $x$ out of $8 x^{2}$ .

$x \cdot x^{2} + x (8 x) + 15 x = 0$

Factor $x$ out of $15 x$ .

$x \cdot x^{2} + x (8 x) + x \cdot 15 = 0$

Factor $x$ out of $x \cdot x^{2} + x (8 x)$ .

$x (x^{2} + 8 x) + x \cdot 15 = 0$

Factor $x$ out of $x (x^{2} + 8 x) + x \cdot 15$ .

$x (x^{2} + 8 x + 15) = 0$

Factor.

Factor $x^{2} + 8 x + 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $8$ .

$3, 5$

Write the factored form using these integers.

$x ((x + 3) (x + 5)) = 0$

Remove unnecessary parentheses.

$x (x + 3) (x + 5) = 0$

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

$x = 0$

$x + 3 = 0$

$x + 5 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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.

$x = - 5$

The final solution is all the values that make $x (x + 3) (x + 5) = 0$ true.

$x = 0, - 3, - 5$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 3, 0), (- 5, 0)$

Step 2

Find the y-intercepts.

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

$y = {(0)}^{3} + 8 {(0)}^{2} + 15 (0)$

Solve the equation.

Remove parentheses.

$y = 0^{3} + 8 {(0)}^{2} + 15 (0)$

Remove parentheses.

$y = 0^{3} + 8 \cdot 0^{2} + 15 (0)$

Remove parentheses.

$y = {(0)}^{3} + 8 {(0)}^{2} + 15 (0)$

Simplify ${(0)}^{3} + 8 {(0)}^{2} + 15 (0)$ .

Simplify each term.

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

$y = 0 + 8 {(0)}^{2} + 15 (0)$

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

$y = 0 + 8 \cdot 0 + 15 (0)$

Multiply $8$ by $0$ .

$y = 0 + 0 + 15 (0)$

Multiply $15$ by $0$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 3, 0), (- 5, 0)$

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

Step 4

Find the x and y Intercepts f(x)=x^3+8x^2+15x

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