Answer:
f(x) = (x + 4)^2*(x - 5)^3
Step-by-step explanation:
For a polynomial P(x) with zeros (or roots):
x₁, x₂, ..., xₙ
And a leading coefficient (the one that multiplies the term of highest degree) A, we can write the polynomial as:
P(x) = A*(x - x₁)*(x - x₂)*...*(x - xₙ)
Now, some of these roots can be repeated.
For example if x₁ = x₂
Then we say that the root x₁ has a multiplicity of two.
And we write the polynomial as:
P(x) = A*(x - x₁)^2*(x - x₃)*....*(x - xₙ)
Now, if we have a polynomial with the roots (or zeros):
Zero at -4 with a multiplicity of 2 (we have the root x = -4 two times)
Zero at 5 with a multiplicity of 3 (we have the root x = 5 3 times)
(And a leading coefficient A = 1, I assume)
This polynomial will be written as:
f(x) = (x - (-4))*(x - (-4))*(x - 5)*(x - 5)*(x - 5)
f(x) = (x + 4)*(x + 4)*(x - 5)*(x - 5)*(x - 5)
f(x) = (x + 4)^2*(x - 5)^3
The correct option is the third one:
