Polynomial Formula

Question

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

Find a possible formula for P(x)P(x). 

Answer 1

P(x) = ax^2(x-5)^2(x+2)

= a(x^2)(x^2-10x+25)(x+2)

= a(x^2)(x^3-8x^2 +5x +50)

= a(x^5 -8x^4 +5x^3 +50x^2)

where a can be any real number, such as 2

= 2x^5 -16x^4 +10x^3 +100x^2

but the problem specified leading coefficient = 1, so a=1

Answer 2

P(x) = x²(x - 5)²(x + 2)