A polynomial function f has a zero of −6, a zero of 4, and a zero of −1 with a multiplicity of 2.

Question

A polynomial function f has a zero of −6, a zero of 4, and a zero of −1 with a multiplicity of 2.
Which equation could represent f?
A) f (x) = x4 + 4x3 − 19x2 − 46x – 24
B) f (x) = x4 − 4x3 − 19x2 + 46x – 24
C) f (x) =2x
4 + 4x3 − 46x
2 + 4x – 48
D) f (x) = - x
4 + 4x3 − 19x2 − 46x – 24

GPT-4o mini · Answer

To find a polynomial function $ f(x) $ that has the given zeros, we can start by constructing the polynomial based on its roots: 1. **Zeros of the function:** - A zero at $ x = -6 $ gives a factor of $ (x + 6) $. - A zero at $ x = 4 $ gives a factor of $ (x - 4) $. - A zero at $ x = -1 $ with a multiplicity of 2 gives a factor of $ (x + 1)^2 $. 2. **The polynomial can be expressed as:** $$ f(x) = k(x + 6)(x - 4)(x + 1)^2 $$ where $ k $ is a constant. 3. **Expanding the polynomial:** To find $ f(x) $, we start by expanding $ (x + 1)^2 $: $$ (x + 1)^2 = x^2 + 2x + 1 $$ 4. **Next, we multiply that by $ (x + 6)(x - 4) $:** $$ (x + 6)(x - 4) = x^2 + 2x - 24 $$ 5. **Now we multiply this result by $ (x^2 + 2x + 1) $:** $$ f(x) = (x^2 + 2x - 24)(x^2 + 2x + 1) $$ 6. **Expanding the resulting polynomial:** - First, distribute $ (x^2 + 2x - 24) $ into $ (x^2 + 2x + 1) $: $$ = x^2(x^2 + 2x + 1) + 2x(x^2 + 2x + 1) - 24(x^2 + 2x + 1) $$ - Which results in: $$ = x^4 + 2x^3 + x^2 + 2x^3 + 4x^2 + 2x - 24x^2 - 48x - 24 $$ - Combine like terms: $$ f(x) = x^4 + 4x^3 - 19x^2 - 46x - 24 $$ 7. **Matching with the given options:** - The polynomial $ f(x) = x^4 + 4x^3 - 19x^2 - 46x - 24 $ matches option (A). Thus, the equation that could represent $ f $ is: **A) $ f(x) = x^4 + 4x^3 - 19x^2 - 46x - 24 $**.