To determine the polynomial function [tex]\( f(x) \)[/tex] with the given zeros and their multiplicities, we will use the Factor Theorem. Let's go through the steps:
### Step-by-Step Solution
1. Identify the given zeros and their multiplicities:
- Zero at [tex]\( x = 4 \)[/tex] with a multiplicity of 1.
- Zero at [tex]\( x = 1 \)[/tex] with a multiplicity of 2.
2. Form the polynomial using the zeros and their multiplicities:
According to the Factor Theorem, if [tex]\( r \)[/tex] is a root of a polynomial [tex]\( f(x) \)[/tex] with a multiplicity [tex]\( m \)[/tex], then [tex]\( (x - r)^m \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
Therefore, given the zeros and their multiplicities:
[tex]\[
f(x) = (x - 4)^1 \cdot (x - 1)^2
\][/tex]
3. Expand the polynomial:
- First, expand [tex]\((x - 1)^2\)[/tex]:
[tex]\[
(x - 1)^2 = (x - 1)(x - 1) = x^2 - 2x + 1
\][/tex]
- Next, multiply by [tex]\((x - 4)\)[/tex]:
[tex]\[
f(x) = (x - 4)(x^2 - 2x + 1)
\][/tex]
- Now, perform the multiplication:
- Distribute [tex]\( x - 4 \)[/tex] across [tex]\( x^2 - 2x + 1 \)[/tex]:
[tex]\[
f(x) = x(x^2 - 2x + 1) - 4(x^2 - 2x + 1)
\][/tex]
- Calculate each term:
[tex]\[
x \cdot (x^2 - 2x + 1) = x^3 - 2x^2 + x
\][/tex]
[tex]\[
-4 \cdot (x^2 - 2x + 1) = -4x^2 + 8x - 4
\][/tex]
- Combine the terms:
[tex]\[
f(x) = x^3 - 2x^2 + x - 4x^2 + 8x - 4
\][/tex]
- Simplify by combining like terms:
[tex]\[
f(x) = x^3 - 6x^2 + 9x - 4
\][/tex]
### Final Result
The polynomial function [tex]\( f(x) \)[/tex] in expanded form is:
[tex]\[
f(x) = x^3 - 6x^2 + 9x - 4
\][/tex]
So, the polynomial function [tex]\( f(x) \)[/tex] that has a zero at [tex]\( 4 \)[/tex] with multiplicity 1 and a zero at [tex]\( 1 \)[/tex] with multiplicity 2 is [tex]\( x^3 - 6x^2 + 9x - 4 \)[/tex].
