Question

Write a polynomial in factored form that has the given zeros:

Zero at -4 with multiplicity of 2
Zero at 5 with multiplicity of 3

Answers to choose from.
1. f(x)=(x-4)^2(x+5)^3
2. f(x)=(x-2)^-4(x+3)^5
3. f(x)=(x+4)^2(x-5)^3
4. f(x)=(x+2)^-4(x+3)^5

Answer 1

f(x) = (x + 4)^2*(x - 5)^3

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: