Division of polynomials

Level 2

If 2 x 3 2x^3 + 4 x 2 4x^2 +2ax+b is exactly divisible by x 2 x^2 +1, then the value of a will be


The answer is 1.

Arpit Sah by Arpit Sah , 22, India

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1 solution

Ayush G Rai
Aug 7, 2016

Since x 2 + 1 x^2+1 is a factor of f ( x ) = 2 x 3 + 4 x 2 + 2 a x + b , x 2 + 1 = 0 ⇒ x = ± − 1 = ± i . f(x)=2x^3+4x^2+2ax+b,x^2+1=0\Rightarrow x=\pm\sqrt{-1}=\pm i.
Case ( 1 ) : (1): When x = + i . x=+i.
2 i 3 + 4 i 2 + 2 a i + b = 0 ⇒ − 2 i − 4 + 2 a i + b = 0. − − − − 1 2i^3+4i^2+2ai+b=0\Rightarrow -2i-4+2ai+b=0.- - - -1
Case ( 2 ) : (2): When x = − i . x=-i.
− 2 i 3 + 4 i 2 − 2 a i + b = 0 ⇒ 2 i − 4 − 2 a i + b = 0. − − − − 2 -2i^3+4i^2-2ai+b=0\Rightarrow 2i-4-2ai+b=0.- - - - 2
We multiply the second equation by − 1 -1 to get − 2 i + 4 + 2 a i − b = 0. − − − − − 3 -2i+4+2ai-b=0.- - - - - 3
Adding equation 1 1 and 3 , 3, we get
− 4 i + 4 a i = 0 ⇒ a = 1 . -4i+4ai=0\Rightarrow a=\boxed 1.


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