write a polynomial function with the given zeros, multiplicities and degrees

Question

1) degree of 3, zeros are -2 multiplicity 2 and -1

2) degree of 4, zeros are 0 multiplicity of 2 and 5 multiplicity 2

3) quartic polynomial, zeros are 3 and 2 multiplicity 3

Answer 1

If a polynomial function has a zero of a, then it has a factor of (x - a). Multiplicity tells you how many times that factor is a factor.

A simple example with just numbers: the prime factorization of 360 = 2³ * 3² * 5.

It's factors are 2 (with multiplicity 3), 3 (with multiplicity 2) and 5.

Looking at number 1 we have:

f(x) = (x - (-2))² (x - (-1))

f(x) = (x + 2)² (x + 1)

The directions do not say that the function needs to be simplified into standard form, but if we do we get:

f(x) = x³ + 5x² + 8x + 4 which clearly is degree 3

Looking at number 2 we have:

f(x) = (x - 0)² (x - 5)²

f(x) = x² (x - 5)²

Can you do number 3 on your own? Scroll down for the final answer . . .

f(x) = (x - 3) (x - 2)³ or f(x) = x⁴ - 9x³ + 30x² - 44x + 24

Answer 2

1) (x+2)^2(x+1)

2) x^2(x-5)^2

3) (x-3)(x-2)^3