A looooooong polynomial!

Algebra Level 3

It is given that : 4 x 4 + a x 3 + b x 2 + 6 x + 1 = [ P ( x ) ] 2 4x^4+ax^3+bx^2+6x+1=[P(x)]^2 ,then find the positive value of a + b a+b .


The answer is 25.

Lorenc Bushi by Lorenc Bushi , 21, Albania

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1 solution

Let P ( x ) = 2 x 2 + m x + 1 P(x)=2x^2+mx+1 . Then, comparing coefficients of like terms on the two sides of the identity, we get

2 m = 6 ⟹ m = 3 , 4 m = a ⟹ a = 12 , m 2 + 4 = b ⟹ b = 13 2m=6\implies m=3, 4m=a\implies a=12, m^2+4=b\implies b=13

So a + b = 12 + 13 = 25 a+b=12+13=\boxed {25}

0 Helpful 0 Interesting 1 Brilliant 0 Confused

We should also consider the case where P(x) takes the form P(x) = 2 x^2 + n x - 1. In this case, a calculation shows a+b < 0, contrary to the fact that we require a positive answer.

Ron Gallagher - 11 months, 3 weeks ago

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