Polynomial Factors: Find a & b | Math Problem

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The polynomial P(x) = x² + ax + 1 is a factor of T(x) = 2x³ - 16x + b, leading to the equations c + 2a = 0, 2 + ac = -16, and b = c. By solving these equations, the values of a are determined to be 3 or -3, resulting in corresponding values of b as -6 or 6. This analysis provides a clear method for finding coefficients in polynomial factorization.

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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?
 
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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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