Find all zeros of the function f(x) = 9x^3 -36x^2-55x+50  Enter the zeros separated by commas.

To find the zeros of the function \( f(x) = 9x^3 - 36x^2 - 55x + 50 \), we will solve for \( x \) where \( f(x) = 0 \). This is a cubic equation, and solving it involves several steps.

### Step 1: Try Rational Root Theorem
The Rational Root Theorem helps us identify possible rational roots. For a polynomial of the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 \), the possible rational roots are the factors of the constant term \( a_0 \) divided by the factors of the leading coefficient \( a_n \).

For the polynomial \( 9x^3 - 36x^2 - 55x + 50 \):
- The constant term \( a_0 = 50 \)
- The leading coefficient \( a_n = 9 \)

The possible rational roots are the factors of 50 divided by the factors of 9. These are:

\[
\text{Possible roots} = \pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{25}{3}, \pm \frac{50}{3}
\]

### Step 2: Test possible roots using synthetic division
Testing some of these roots using synthetic division, we find that \( x = 5 \) is a root.

### Step 3: Factor the cubic polynomial
Since \( x = 5 \) is a root, we can factor \( f(x) \) as \( (x - 5) \) times a quadratic. We can perform synthetic division of \( 9x^3 - 36x^2 - 55x + 50 \) by \( x - 5 \).

After performing synthetic division, the quotient is \( 9x^2 + 9x - 10 \).

So, the factored form of \( f(x) \) is:

\[
f(x) = (x - 5)(9x^2 + 9x - 10)
\]

### Step 4: Solve the quadratic equation
Now, we solve \( 9x^2 + 9x - 10 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 9 \), \( b = 9 \), and \( c = -10 \).

\[
x = \frac{-9 \pm \sqrt{9^2 - 4(9)(-10)}}{2(9)}
\]
\[
x = \frac{-9 \pm \sqrt{81 + 360}}{18}
\]
\[
x = \frac{-9 \pm \sqrt{441}}{18}
\]
\[
x = \frac{-9 \pm 21}{18}
\]

Thus, we get two solutions:

\[
x = \frac{-9 + 21}{18} = \frac{12}{18} = \frac{2}{3}, \quad x = \frac{-9 - 21}{18} = \frac{-30}{18} = -\frac{5}{3}
\]

### Step 5: List all zeros
The zeros of the function are:
\[
x = 5, \, x = \frac{2}{3}, \, x = -\frac{5}{3}
\]

Thus, the zeros of the function \( f(x) = 9x^3 - 36x^2 - 55x + 50 \) are:
\[
5, \frac{2}{3}, -\frac{5}{3}
\]

Do you want more details on any step? Here are some related questions you can explore:
1. How do you use synthetic division to factor polynomials?
2. Can you explain the Rational Root Theorem in more detail?
3. What are the steps to solve a quadratic equation using the quadratic formula?
4. How can you graph the cubic function to verify its zeros?
5. What are some other methods to solve cubic equations?

**Tip**: Always check the discriminant in a quadratic equation to determine the nature of the roots (real vs. complex).

Find all zeros of the function f(x) = 9x^3 - 36x^2 - 55x + 50. Enter the zeros separated by commas.

Learn how to find all zeros of the cubic function f(x) = 9x^3 - 36x^2 - 55x + 50. This step-by-step guide includes using the Rational Root Theorem and solving a quadratic equation to find the roots. Suitable for students in Grades 10-12.

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