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Jaime was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write? zero:x=4,x=-2 multiplicity:1,3

a. f(x)=(x+4)(x-2)
b. f(x)=(x-4)(x+2)
c. f(x)=(x-4)(x+2)^3
d. f(x)=(x+4)(x-2)^3

1 answer

To construct a polynomial function from the given zeros and their multiplicities, we need to follow these steps:

  1. Identify the zeros and their multiplicities:

    • Zero \(x = 4\) with multiplicity \(1\)
    • Zero \(x = -2\) with multiplicity \(3\)

  2. Write the polynomial as a product of factors based on these zeros:

    • The factor for the zero \(x = 4\) is \((x - 4)\)
    • The factor for the zero \(x = -2\) is \((x + 2)\), and since it has a multiplicity of \(3\), it will be raised to the third power: \((x + 2)^3\)

  3. Combine these factors into one polynomial: \[ f(x) = (x - 4)(x + 2)^3 \]

Now, let's compare this with the answer choices:

a. \( f(x) = (x + 4)(x - 2) \) - Not correct. b. \( f(x) = (x - 4)(x + 2) \) - Not correct (missing the third power). c. \( f(x) = (x - 4)(x + 2)^3 \) - Correct. d. \( f(x) = (x + 4)(x - 2)^3 \) - Not correct.

Therefore, the correct function Jaime should write is:

c. \( f(x) = (x - 4)(x + 2)^3 \).