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Question
Factorise:
3x3 – x2 – 3x + 1
Solution
Let p(x) = 3x3 – x2 – 3x + 1
Constant term of p(x) = 1
Factor of 1 are ±1
By trial, we find that p(1) = 0, so (x – 1) is a factor of p(x)
Now, we see that 3x3 – x2 – 3x + 1
= 3x3 – 3x2 + 2x2 – 2x – x + 1
= 3x2(x – 1) + 2x(x – 1) – 1(x – 1)
= (x – 1)(3x2 + 2x – 1)
Now, (3x2 + 2x – 1) = 3x2 + 3x – x – 1 ...[By splitting middle term]
= 3x(x + 1) – 1(x + 1)
= (x + 1)(3x – 1)
∴ 3x3 – x2 – 3x + 1 = (x – 1)(x + 1)(3x – 1)
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