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  • Without actual division, prove that 2x^{4} – 5x^{3} + 2x^{2} – x + 2 is divisible by x^{2} – 3x + 2.[Hint: Factorize x^{2} – 3x + 2]

    Without actual division, prove that 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 3x + 2. [Hint: Factorize x2–3x + 2]

Answers (1)

First of all, factorize x2 – 3x + 2  

We get

x2 – 2x-3x + 2

= x(x-2)-1(x-2)

= (x-2)(x-1)

According to remainder theorem when p(x)  is divided by (x+a)  then the remainder is p(-a) .

So if p(x) is divide by x2 – 3x + 2 then p(2) and p(1) must be zero

p(2)=2(2)4 -5(2)3 +2(2)2 -2+2

=32-40+8

=0

p(1)=2(1)4 -5(1)3 +2(1)2 -1+2

    =2-5+2-1+2

    =6-6

     =0

Hence both p(1) and p(2) are zero therefore p(x) is divisible by x2 – 3x + 2

Hence proved.

Posted by

infoexpert24

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