The polynomial of degree 5

Question

The polynomial of degree 5,  has leading coefficient 1, has roots of multiplicity 2 at  and , and a root of multiplicity 1 at Find a possible formula for .

Answer 1

For this type of problem, use the factored form of a polynomial of degree 5. For a degree 5 polynomial, there are 5 factors:

P(x) = a·(x-p) (x-q ) (x-r) (x-s) (x-t)

where p, q, r, s, t are the roots and a is the leading coefficient. The problem statement tells us that a = 1. There are 5 roots. The problem statement tells us what they are. A multiplicity of 2 means that the roots occur twice, so we have roots at x=1, x=1, x=0, x=0. A multiplicity of 1 means that the root appears once, so the final root is x = -5. So our equation is:

P(x) = 1·(x-1) (x-1) (x-0) (x-0) (x-(-5)) = (x-1)² x² (x+5)