Find the x and y Intercepts -3x^3+24x^2-45x

$- 3 x^{3} + 24 x^{2} - 45 x$

Step 1

Write $- 3 x^{3} + 24 x^{2} - 45 x$ as an equation.

$y = - 3 x^{3} + 24 x^{2} - 45 x$

Step 2

Find the x-intercepts.

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

$0 = - 3 x^{3} + 24 x^{2} - 45 x$

Solve the equation.

Rewrite the equation as $- 3 x^{3} + 24 x^{2} - 45 x = 0$ .

$- 3 x^{3} + 24 x^{2} - 45 x = 0$

Factor the left side of the equation.

Factor $- 3 x$ out of $- 3 x^{3} + 24 x^{2} - 45 x$ .

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

$- 3 x \cdot x^{2} + 24 x^{2} - 45 x = 0$

Factor $- 3 x$ out of $24 x^{2}$ .

$- 3 x \cdot x^{2} - 3 x (- 8 x) - 45 x = 0$

Factor $- 3 x$ out of $- 45 x$ .

$- 3 x \cdot x^{2} - 3 x (- 8 x) - 3 x \cdot 15 = 0$

Factor $- 3 x$ out of $- 3 x (x^{2}) - 3 x (- 8 x)$ .

$- 3 x (x^{2} - 8 x) - 3 x \cdot 15 = 0$

Factor $- 3 x$ out of $- 3 x (x^{2} - 8 x) - 3 x (15)$ .

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

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 $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x = 0$

$x - 5 = 0$

$x - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

$x = 0, 5, 3$

x-intercept(s) in point form.

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

Step 3

Find the y-intercepts.

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

$y = - 3 {(0)}^{3} + 24 {(0)}^{2} - 45 \cdot 0$

Solve the equation.

Remove parentheses.

$y = - 3 \cdot 0^{3} + 24 {(0)}^{2} - 45 \cdot 0$

Remove parentheses.

$y = - 3 \cdot 0^{3} + 24 \cdot 0^{2} - 45 \cdot 0$

Remove parentheses.

$y = - 3 {(0)}^{3} + 24 {(0)}^{2} - 45 \cdot 0$

Simplify $- 3 {(0)}^{3} + 24 {(0)}^{2} - 45 \cdot 0$ .

Simplify each term.

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

$y = - 3 \cdot 0 + 24 {(0)}^{2} - 45 \cdot 0$

Multiply $- 3$ by $0$ .

$y = 0 + 24 {(0)}^{2} - 45 \cdot 0$

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

$y = 0 + 24 \cdot 0 - 45 \cdot 0$

Multiply $24$ by $0$ .

$y = 0 + 0 - 45 \cdot 0$

Multiply $- 45$ 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 4

List the intersections.

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

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

Step 5

Find the x and y Intercepts -3x^3+24x^2-45x

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