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Question
Using the Factor Theorem, show that (x – 2) is a factor of x3 – 2x2 – 9x + 18. Hence, factorise the expression x3 – 2x2 – 9x + 18 completely.
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Solution
Let f(x)= x3 – 2x2 – 9x + 18
x – 2 = 0 `\implies` x = 2
∴ Remainder = f(2)
= (2)3 – 2(2)2 – 9(2) + 18
= 8 – 8 – 18 +18
= 0
Hence, (x – 2) is a factor of f(x).
Now, we have:
x2 – 9
`x - 2")"overline(x^3 - 2x^2 - 9x + 18)`
x3 – 2x2
– +
– 9x + 18
– 9x + 18
+ –
0
∴ x3 – 2x2 – 9x + 18 = (x – 2)(x2 – 9)
= (x – 2)(x + 3)(x – 3)
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