Polynomial Factors: Find a & b | Math Problem

[evanbirch]

This problem came up on a math contest the other day and I drew a blank - now it's bugging me.

"If the polynomial P(x)=x^2+ax+1 is a factor of T(x)=2x^3-16x+b, find the values of a and b."

Anyone have any ideas?

 

[DeadWolfe]

We have that there is a c such that (x^2+ax+1)(2x+c)=2x^3-16x+b. Thus comparing coefficients gives us:

c+2a=0
2+ac=-16
b=c

Solving this yields

a=3 or -3, and thus
b=-6 or 6