Question

find the fourth degree polynomial with real coefficients having zeros: 0,1,2

Answer 1

If the polynomial has real zeroes 0,1,2 then

x = 0
x = 1
x = 2

Placing all terms on the left side you have

x - 0 = 0
x - 1 = 0
x - 2 = 0

By zero product property

(x - 0)(x - 1)(x - 2) = 0

Since we need a fourth degree polynomial, add another factor:
(x - 0)(x - 1)(x - 1)(x - 2) = 0
x(x - 1)^2(x - 2) = 0

Answer 2

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

Multiply that out to get your fourth degree polynomial.

Answer 3

thank you so much!

Answer 4

You're welcome.