Identify the multiplicities of the linear factors off (x)=(x-1)^(2)(x+4)(2x+5)

Answer 1

To identify the multiplicities of the linear factors of the function \( f(x) = (x-1)^2 (x+4)(2x+5) \), we start by recognizing the linear factors in the expression.

1. The first factor is \( (x-1)^2 \):

   • The linear factor is \( x - 1 \).
   • The exponent is 2, which means the multiplicity is 2.

2. The second factor is \( (x + 4) \):

   • The linear factor is \( x + 4 \).
   • The exponent is 1, which means the multiplicity is 1.

3. The third factor is \( (2x + 5) \):

   • The linear factor is \( 2x + 5 \).
   • This can be rewritten as \( 2(x + \frac{5}{2}) \), but since we are interested in linear factors, we consider it as a factor of degree 1. The exponent is 1, which means the multiplicity is 1.

To summarize, the linear factors and their corresponding multiplicities are:

• \( x - 1 \): multiplicity 2
• \( x + 4 \): multiplicity 1
• \( 2x + 5 \): multiplicity 1

So the final answer is:

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