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if x^4+8x^2+12 = k(x^2 + a )(x^2+2) + 4(x^2 + 2) then find the value of k+a where k and a both are positive integers

if x^4+8x^2+12 = k(x^2 + a )(x^2+2) + 4(x^2 + 2) then find the value of k+a where k and a both are positive integers

Grade:9

2 Answers

Deepak Kumar Shringi
askIITians Faculty 4404 Points
6 years ago
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Samyak Jain
333 Points
5 years ago
x4 + 8x2 + 12 = k(x2 + a) (x2 + 2) + 4(x2 + 2)               [given]
LHS  =  x4 + 6x2 +2x2 + 12  =  x2(x2 + 6)  + 2(x2 + 6)
           = (x2 + 6)(x2 + 2) 
RHS   =  k(x2 + a) (x2 + 2) + 4(x2 + 2) 
          = (kx2 + ka) (x2 + 2) + 4(x2 + 2) 
          = (x2 + 2) (kx2 + ka + 4)
Comparing LHS & RHS, we get kx2 + ka + 4 = x2 + 6
\therefore k = 1 & a = 2   

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