Question

actual division, prove that 2x ^ 4 - 5x ^ 3 + 2x ^ 2 - x + 2 is exactly divisible byx- x ^ 2 - 3x + 2

Answer 1

Step-by-step explanation:

Let p(x) = 2x^4 – 5x^3 + 2x^2 – x+ 2 firstly, factorise x^2-3x+2.

Now, x^2-3x+2 = x^2-2x-x+2 [by splitting middle term]

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

Hence, 0 of x^2-3x+2 are land 2.

We have to prove that, 2x^4 – 5x^3 + 2x^2 – x+ 2 is divisible by x^2-3x+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.

Hope will be helpful ☺️