Bot GPT 4-o mini To construct a polynomial function given the zeros and their multiplicities, we use the fact that each zero \( r \) with multiplicity \( m \) contributes a factor of \( (x - r)^m \) to the polynomial.
From the table given:
- The zero \( x = 4 \) has a multiplicity of 1. Therefore, the factor contributed by this zero is \( (x - 4)^1 = (x - 4) \).
- The zero \( x = -2 \) has a multiplicity of 3. Therefore, the factor contributed by this zero is \( (x + 2)^3 \).
Putting these factors together, the polynomial function can be expressed as:
\[ f(x) = (x - 4)(x + 2)^3 \]
Now, let's match this with the provided options:
A. \( f(x) = (x-4)(x+2)^3 \) ✅
B. \( f(x) = (x+4)(x-2) \)
C. \( f(x) = (x-4)(x+2) \)
D. \( f(x) = (x+4)(x-2)^3 \)
The correct answer is:
A. \( f(x) = (x-4)(x+2)^3 \).
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