Question

1) without actual division , prove that 2x^4-5x^3+2x^2-x+2 is exactly divisible by x^2-3x+2.
2) without doing actual division , prove that x^4+2x^3-2x^2+2x-3 is exactly divisible by x^2+2x-3

Answer 1

Here is ur answer:

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.

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