The polynomial of degree 5, P ( x ) has leading coefficient 1, has roots of multiplicity 2 at x = 5 and x = 0 , and a root of multiplicity 1 at x = − 2

Question

Find the formula

P(x)=

Answer 1

If P(x) has roots of 5, 5, 0, and -2 then it has factors of (x - 5), (x - 5), (x - 0), and (x + 2) so can be written as:

P(x) = a(x - 5)(x - 5)(x - 0)(x + 2)

In this case, we are told (in so many words) that a = 1 so

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

P(x) = x(x - 5)(x - 5)(x + 2) or

P(x) = x(x - 5)²(x + 2)