https://file.aiquickdraw.com/mathbot/file/17294690128874n61ys5l.jpg

The problem asks for the zeros of the polynomial function:

\[
f(x) = 6(x - 9)(x + 4)^2
\]

### Steps to find the zeros and their multiplicities:

1. **Set \( f(x) = 0 \) to find the zeros:**
   \[
   6(x - 9)(x + 4)^2 = 0
   \]
   This equation is satisfied if either factor equals zero.

2. **Solve for each factor:**
   - For \( x - 9 = 0 \), the solution is \( x = 9 \). This zero has multiplicity **1** (because the exponent on this factor is 1).
   - For \( (x + 4)^2 = 0 \), the solution is \( x = -4 \). This zero has multiplicity **2** (because the exponent is 2).

### Zeros and their multiplicities:
- Zero at \( x = 9 \), with multiplicity 1.
- Zero at \( x = -4 \), with multiplicity 2.

### Behavior at the x-axis:
- At \( x = 9 \), since the multiplicity is 1 (odd), the graph **crosses** the x-axis.
- At \( x = -4 \), since the multiplicity is 2 (even), the graph **touches** the x-axis and **turns around**.

### Final answer:
The zeros are \( x = 9 \) and \( x = -4 \).

Would you like more details or explanations on any step? Here are some related questions:

1. What is the general relationship between multiplicity and the graph's behavior at zeros?
2. How do higher-order multiplicities (greater than 2) affect the graph at the x-axis?
3. Can you always factor polynomials like this easily?
4. What happens if a polynomial has complex roots?
5. How does the degree of the polynomial affect the number of zeros?

**Tip:** The graph crosses the x-axis at zeros with odd multiplicity and touches (but does not cross) at zeros with even multiplicity!

Find the zeros for the polynomial function f(x) = 6(x - 9)(x + 4)^2 and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

Explore how to find the zeros and multiplicities of the polynomial function f(x) = 6(x - 9)(x + 4)^2. Learn how the graph behaves at the zeros based on their multiplicities, including when it crosses or touches the x-axis. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation