Question

indentify the multiplicities of the linear f(x)=\left(x-1\right)^2\left(x+4\right)\left(2x+5\right)

Answer 1

To identify the multiplicities of the linear factors of the polynomial \( f(x) = (x-1)^2 (x+4) (2x+5) \), we can analyze each factor.

1. **Factor \( (x-1)^2 \)**:
- This factor indicates that the root \( x = 1 \) has a multiplicity of 2, since the exponent is 2.

2. **Factor \( (x+4) \)**:
- This factor indicates that the root \( x = -4 \) has a multiplicity of 1, since the exponent is 1.

3. **Factor \( (2x+5) \)**:
- To find the root of this factor, we set it equal to zero:
\[
2x + 5 = 0 \implies 2x = -5 \implies x = -\frac{5}{2}
\]
- Since this is a linear factor (the exponent is 1), the root \( x = -\frac{5}{2} \) also has a multiplicity of 1.

### Summary of Multiplicities:
- \( x = 1 \): multiplicity 2
- \( x = -4 \): multiplicity 1
- \( x = -\frac{5}{2} \): multiplicity 1

Thus, the multiplicities of the roots of the polynomial \( f(x) \) are:
- \( x = 1 \) has a multiplicity of 2,
- \( x = -4 \) has a multiplicity of 1,
- \( x = -\frac{5}{2} \) has a multiplicity of 1.