Question

without actual division, prove that 2x^4-5x^3+2x^2-x+2 is divisible by x^2-3x+2 .​

Answer 1

Answer:

First of all,

x^2-3x+2

=>x^2-2x-1x+2

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

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

Therefore,(x-2)(x-1)are the factors.

i) (x-2)

x-2=0

=>x=2

So,p(x)=2

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

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

=32-40+8

= -40+40=0

Hence,it proves that (x-2) is a factor .

ii) (x-1)

=>x-1=0

=>x=1

So,p(x)=1

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

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

=2-5+2-1+2

=6-6=0

Hence,it proves that (x-1) is a factor.

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