Question

without actual division prove that 2x^4-6x^3+3x^2+3x-2 is exactly divisible by x^2
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Answer 1

Answer:

x²-3x+2

Step-by-step explanation:

Given f(x) = 2x⁴-6x³+3x²+3x-2

x²-3x+2 = x²-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 x²-3x+2.

If f(1) = 0 and f(2) = 0, then f(x) is exactly divisible by x²-3x+2.

f(x) = 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) and (2), we get

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

Therefore, f(x) is exactly divisible by x²-3x+2.