Question

indentify the multiplications of the linear factor h(x)=\left(x+4\right)\left(x-1\right)^2\left(x+3\right)^5

Answer 1

To identify the multiplicities of the linear factors in the polynomial \( h(x) = (x+4)(x-1)^2(x+3)^5 \), we look at each linear factor and their corresponding exponents:

1. **Factor \( (x + 4) \)**:
- This factor has an exponent of \( 1 \).
- **Multiplicity**: \( 1 \)

2. **Factor \( (x - 1)^2 \)**:
- This factor has an exponent of \( 2 \).
- **Multiplicity**: \( 2 \)

3. **Factor \( (x + 3)^5 \)**:
- This factor has an exponent of \( 5 \).
- **Multiplicity**: \( 5 \)

Now we can summarize the multiplications of the linear factors:

- \( x + 4 \) has a multiplicity of \( 1 \)
- \( x - 1 \) has a multiplicity of \( 2 \)
- \( x + 3 \) has a multiplicity of \( 5 \)

Thus, the complete list of linear factors and their multiplicities in \( h(x) \) is:

- \( (x + 4) \): Multiplicity \( 1 \)
- \( (x - 1) \): Multiplicity \( 2 \)
- \( (x + 3) \): Multiplicity \( 5 \)