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Question
Using the Factor Theorem, show that (3x + 2) is a factor of 3x3 + 2x2 – 3x – 2. Hence, factorise the expression 3x3 + 2x2 – 3x – 2 completely.
Solution
Let f(x) = 3x3 + 2x2 – 3x – 2
3x + 2 = 0 `\implies x = (-2)/3`
∴ Remainder = `f ((-2)/3)`
= `3((-2)/3)^3 + 2((-2)/3)^2 - 3((-2)/3) - 2`
= `(-8)/9 + 8/9 + 2 - 2`
= 0
Hence, (3x + 2) is a factor of f(x).
Now, we have:
x2 – 1
`3x + 2")"overline(3x^3 + 2x^2 - 3x - 2)`
3x3 + 2x2
– –
– 3x – 2
– 3x – 2
+ +
0
∴ 3x3 + 2x2 – 3x – 2 = (3x + 2)(x2 – 1)
= (3x + 2)(x + 1)(x – 1)
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