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Question

Without actual division, prove that 2x4 − 6x3 + 3x2 + 3x − 2 is exactly divisible by x2 − 3x + 2.

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Solution

Let fx=2x4-6x3+3x2+3x-2 and gx=x2-3x+2.
gx=x2-3x+2 =x2-2x-x+2 =xx-2-1x-2 =x-1x-2
Thus, f(x) is divisible by g(x) if f(1) = 0 and f(2) = 0. Now
f1=214-613+312+31-2=2-6+3+3-2=0f2=224-623+322+32-2=32-48+12+6-2=0
Hence, 2x4 − 6x3 + 3x2 + 3x − 2 is exactly divisible by x2 − 3x + 2.

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