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Question

Without actual division , prove that 2x45x3+2x2x+2 is divisible by x23x+2.

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Solution


Let p(x)=2x45x3+2x2x+2
Factorise x23x+2.
Now, x23x+2=x22xx+2 [By splitting middle term]
=x(x2)1(x2)=(x1)(x2)
Hence, zeros of x23x+2 are 1 and 2.
We have to prove that, 2x45x3+2x2x+2 is divisible by x23x+2.
i.e. prove that p(1)=0 and p(2)=0
Now. p(1)=2(1)45(1)3+2(1)21+2
=25+21+2=66=0
p(2)=2(2)45(2)3+2(2)22+2
=2×165×8+2×4+0
=3240+8=4040=0
Hence, p(x) is divisible by x23x+2.

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