Question

Without actual division ,proovd that 2x^4-8x^3+3x^2+12x-9 is actually divisible by x^2-4x+3?

Answer 1

Let f(x) = 2x^4 - 8x^3 + 3x^2 +12 -9
if f(x) is exactly divisible by x^2 -* 4x +3 = (x-1) (x-3) then f(1)=0 and f(3)=0
thus,
f(1)= 2(1)^4 - 8(1)^3 3(1)^2 +12(1) -9
= 2 - 8 +3 +12 -9
=0
and,
f(3)= 2(3)^4 -8(3)^3 + 3(3)^2 + 12(3) -9
=162 - 216 +27 + 36 - 9
=0
hence f(x) is exactly divisible by x^2 - 4x +3