Question

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

Answer 1

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.



hope it may help you.....

Answer 2

Let p(x) = 2x4 – 5x3 + 2x2 – x+ 2 firstly, factorise x2-3x+2.
Now, x2-3x+2 = x2-2x-x+2 [by splitting middle term]
= x(x-2)-1 (x-2)= (x-1)(x-2)
Hence, 0 of x2-3x+2 are land 2.
We have to prove that, 2x4 – 5x3 + 2x2 – x+ 2 is divisible by x2-3x+2 i.e., to prove that p (1) =0 and p(2) =0
Now, p(1) = 2(1)4 – 5(1)3 + 2(1)2 -1 + 2 =2-5+2-1+2=6-6=0
and p(2) = 2(2)4 – 5(2)3 + 2(2)2 – 2 + 2 = 2x16-5x8+2x4+ 0 = 32 – 40 + 8 = 40 – 40 =0
Hence, p(x) is divisible by x2-3x+2.