Find a polynomial function​ f(x) of degree 3 with real coefficients that satisfies the following conditions. Zero of 0 and zero of 1 having multiplicity​ 2; ​f(​2)=10

Question

The polynomial function is f(x)=

Answer 1

ax(x-1)^2

f(2)=10

10=a(2)(2-1)^2 = 2a

a =10/2 =5

f(x) = 5(x)(x-1)^2 = 5x^3 -10x^2 +5x is a 3 degree polynomial with zeroes 0 and 1 and f(2)=10

Answer 2

Zero of 0 means x is a factor of the polynomial function.

Zero of 1 means (x - 1) is a factor of the polynomials function, and since the multiplicity for this is 2 there are two (x - 1) factors.

Let f(x) = Ax(x - 1)²

f(2) = A(2)(2 - 1)(2 - 1) = 10

2A = 10

A = 5

f(x) = 5x(x - 1)(x - 1)

f(x) = 5x^3 - 10x^2 + 5x.