Polynomial formula

Question

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

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

Answer 1

General form of polynomial = leading coefficent * (x- root1)^(root 1 multiplicity) * (x-root2)^(root 2 multiplicity) .....

Given that leading coefficient = 1.

Root 1 = 1 with multiplicity 2 . so factor will become (x-1)^2

Root 2 = 0 with multiplicity 2 . so factor will become (x-0)^2 = x^2

Root 3 = -1 with multiplicity 1 . so factor will become (x+1)

So polynomial = x^2. (x+1) . (x-1)^2

Answer 2

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

= x^2(x^2-2x+1)(x+1)

= x^2 (x^3 -x^2 -x +1)

= x^5 -x^4 -x^3 + x^2

take each zero and change its sign, then stick an x in front of it to get a factor

if there's multiplicity two, stick an exponent of 2 for the factor

then multiply them all together. Carefully, or it's easy to make a mistake.