Polynomial Graphs

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) = (x-5)^2(x)^2(x+2) = (x^2-10x+ 25)(x^2)(x+2) = (x^3- 8x^2 + 5x +50)(x^2)

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

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

x=5

x-5 if it's multiplicity 2, put 2 as an exponent over the factor: (x-5)^2

x=-2

x+2= 0 (x+2) has multiplicity 1,

x= 0

x^2 It's multiplicity 2

multiply them all together (x-5)^2(x+2)(x^2) It's 5 degree polynomial