Greg was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write?
Zero Multiplicity
x=−4
2
x=2
3
Function #1: f(x)=(x+4)(x−2)
Function #2: f(x)=(x+4)2(x−2)3
Function #3: f(x)=(x−4)2(x+2)3
Function #2: f(x)=(x+4)2(x-2)3
Function #2: f(x)=(x+4)2(x−2)3.
To construct a polynomial function with given zeros and multiplicities, we need to determine the factors of the polynomial.
In this case, we are given two zeros: x = -4 with multiplicity 2 and x = 2 with multiplicity 3.
To write the polynomial function, we need to use the factors (x+a)^b, where 'a' is the zero and 'b' is the multiplicity.
Let's break down the given options:
Function #1: f(x) = (x+4)(x-2)
This function only accounts for the zeros x = -4 and x = 2, but it does not consider the multiplicities.
Function #2: f(x) = (x+4)^2(x-2)^3
In this function, we have accounted for the multiplicities. (x+4)^2 indicates that x = -4 is a zero with multiplicity 2, and (x-2)^3 indicates that x = 2 is a zero with multiplicity 3. This is the correct function that Greg should write.
Function #3: f(x) = (x-4)^2(x+2)^3
This function does not represent the given zeros and their multiplicities accurately. It has x = 4 as a zero with multiplicity 2, instead of x = -4, and x = -2 as a zero with multiplicity 3, instead of x = 2.
Therefore, the function Greg should write is Function #2: f(x) = (x+4)^2(x-2)^3.