Question
indentify the multiplicites of the linear factors of h(x) = (x + 4) (x - 1)^2 (x + 3)^5
Answers
GPT-4o mini
To identify the multiplicities of the linear factors of the polynomial \( h(x) = (x + 4)(x - 1)^2(x + 3)^5 \), we can look at each factor individually:
1. \( (x + 4) \): This factor is linear and appears once. Therefore, its multiplicity is **1**.
2. \( (x - 1)^2 \): This factor is also linear but appears squared. Thus, its multiplicity is **2**.
3. \( (x + 3)^5 \): This factor is linear and appears raised to the fifth power. Therefore, its multiplicity is **5**.
In summary, the multiplicities of the linear factors of \( h(x) \) are:
- For \( x + 4 \): multiplicity **1**
- For \( x - 1 \): multiplicity **2**
- For \( x + 3 \): multiplicity **5**
1. \( (x + 4) \): This factor is linear and appears once. Therefore, its multiplicity is **1**.
2. \( (x - 1)^2 \): This factor is also linear but appears squared. Thus, its multiplicity is **2**.
3. \( (x + 3)^5 \): This factor is linear and appears raised to the fifth power. Therefore, its multiplicity is **5**.
In summary, the multiplicities of the linear factors of \( h(x) \) are:
- For \( x + 4 \): multiplicity **1**
- For \( x - 1 \): multiplicity **2**
- For \( x + 3 \): multiplicity **5**