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

Answer 1

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) \( f(x) = x^4 + 4x^3 - 19x^2 - 46x - 24 \)**.