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Question

Without actual division prove that x4+2x32x2+2x3 is exactly divisible by x2+2x3.

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Solution

To prove :
x4+2x32x2+2x3 is exactly divisible by x2+2x3

Proof :
Let, p(x)=x4+2x32x2+2x3
And, g(x)=x2+2x3

Then, g(x)=x2+2x3
=x2+3xx3=x(x+3)(x+3)=(x+3)(x1)

Now, we check if g(x) is a factor of p(x) by using factor theorem.

(x+3) and (x1) divides p(x) ifp(3) and p(1)=0
So,
p(3)=(3)4+2(3)32(3)2+2(3)3
=81541863=0
and,
p(1)=(1)4+2(1)32(1)2+2(1)3
=1+22+23=0

Hence, p(x) is divisible by (x+3) and (x1)
p(x) is divisible by (x+3)(x1)
p(x) is divisible by g(x)
Hence proved.

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