Question

Prove that 2x⁴-6x³+3x²+3x+x-2 is exatly divisible by x²-3x+2 by actual division​

Answer 1

Given f(x) = 2x^4 - 6x^3 + 3x^2 + 3x - 2

x^2 - 3x + 2 = x^2 - x - 2x + 2

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

= (x - 1)(x - 2)

If (x - 1) and (x - 2) are the factors of f(x).Then f(x) is divisible by x^2 - 3x + 2.

if f(1) = 0 and f(2) = 0, then f(x) is exactly divisible by x^2 - 3x + 2.

f(1) = 2(1)^4 - 6(1)^3 + 3(1)^2 + 3(1) - 2

= 2 - 6 + 3 + 3 - 2

= 0. -------- (1)

f(2) = 2(2)^4 - 6(2)^3 + 3(2)^2 + 3(2) - 2

= 2(16) - 6(8) + 3(4) + 6 - 2

= 32 - 48 + 12 + 6 - 2

= 0. ------- (2)

From (1) & (2), we get

f(1) = 0 and f(2) = 0.

Therefore f(x) is exactly divisible by x^2 - 3x + 2.