Polynomial Factors

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

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evanbirch	Polynomial Factors Tweet 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