Question

A polynomial g(x) is given by:
g(x) = (x-1)(x+2)(x-a)(x-b)

1) Given that g(0)=4 show that ab=-2

2) given also that g(3)=40 show that a+b=1

3) given that a and b are integers, give possible values for a and b.

Thank you

Answer 1

g(0) = (-1)(2)(-a)(-b) = -2ab

g(3) = (2)(5)(3-a)(3-b)
(3-a)(3-b) = 4
9-3a-3b+ab = 4
9-3(a+b)-2 = 4
-3(a+b) = -3
a+b = 1

(a+b)^2 = a^2+b^2+2ab = a^2+b^2-4 = 1
a^2+b^2 = 5

Now it should be clear what a and b are.