Activity 2: T'S YOUR TURN

Directions: Identify the multiplicity of the root/zero of the polynomial function. Write

your answers on a separate sheet of paper.

Polynomial Function

1.y= (x+1)⁵(x-1)³

2. y= (x+4)²(x-3)³

3. y= x(x-3)⁴(x+4)⁶

Root/Zero

1.

2.

3.

Multiplicity

1.

2.

3.

Let's identify the roots and their multiplicities for each polynomial function.

### 1. \( y = (x + 1)^5(x - 1)^3 \)

**Root/Zero:**

- The root for \( (x + 1)^5 \) is \( x = -1 \).
- The root for \( (x - 1)^3 \) is \( x = 1 \).

**Multiplicity:**

- The root \( x = -1 \) has multiplicity 5 because of the exponent on \( (x + 1) \).
- The root \( x = 1 \) has multiplicity 3 because of the exponent on \( (x - 1) \).

### 2. \( y = (x + 4)^2(x - 3)^3 \)

**Root/Zero:**

- The root for \( (x + 4)^2 \) is \( x = -4 \).
- The root for \( (x - 3)^3 \) is \( x = 3 \).

**Multiplicity:**

- The root \( x = -4 \) has multiplicity 2.
- The root \( x = 3 \) has multiplicity 3.

### 3. \( y = x(x - 3)^4(x + 4)^6 \)

**Root/Zero:**

- The root for \( x \) is \( x = 0 \).
- The root for \( (x - 3)^4 \) is \( x = 3 \).
- The root for \( (x + 4)^6 \) is \( x = -4 \).

**Multiplicity:**

- The root \( x = 0 \) has multiplicity 1 (since there is no exponent).
- The root \( x = 3 \) has multiplicity 4.
- The root \( x = -4 \) has multiplicity 6.

---

### Summary:

1. **Root/Zero:** \( x = -1 \), \( x = 1 \)  
   **Multiplicity:** 5, 3

2. **Root/Zero:** \( x = -4 \), \( x = 3 \)  
   **Multiplicity:** 2, 3

3. **Root/Zero:** \( x = 0 \), \( x = 3 \), \( x = -4 \)  
   **Multiplicity:** 1, 4, 6

Would you like more details on this process or have any questions?

### Related Questions:
1. How do you determine if a polynomial has real or complex roots?
2. What does it mean when a root has a higher multiplicity?
3. How does the graph of a polynomial behave near roots with different multiplicities?
4. Can a polynomial have a root with multiplicity 0? Why or why not?
5. How does the degree of a polynomial relate to the number of roots (counting multiplicity)?

### Tip:  
Roots with odd multiplicities (e.g., 1, 3, 5) will cross the x-axis, while roots with even multiplicities (e.g., 2, 4, 6) will "bounce" off the x-axis.

Activity 2: IT'S YOUR TURN

Directions: Identify the multiplicity of the root/zero of the polynomial function. Write your answers on a separate sheet of paper.

Polynomial Function
1. y= (x+1)⁵(x-1)³
2. y= (x+4)²(x-3)³
3. y= x(x-3)⁴(x+4)⁶

Root/Zero
1.
2.
3.

Multiplicity
1.
2.
3.

Learn how to identify the multiplicities of the roots of polynomial functions such as y= (x+1)⁵(x-1)³, y= (x+4)²(x-3)³, and y= x(x-3)⁴(x+4)⁶. This guide explains key concepts like roots/zeros and multiplicity, and provides a step-by-step solution. Suitable for students in Grades 9-12.

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