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Question
Without actual division, prove that 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 3x + 2. [Hint: Factorise x2 – 3x + 2]
Solution
Let p(x) = 2x4 – 5x3 + 2x2 – x + 2 firstly, factorise x2 – 3x + 2.
Now, x2 – 3x + 2 = x2 – 2x – x + 2 ...[By splitting middle term]
= x(x – 2) – 1(x – 2) = (x – 1)(x – 2)
Hence, 0 of x2 – 3x + 2 are 1 and 2.
We have to prove that, 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 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.
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