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Question
Factorise the expression f(x) = 2x3 – 7x2 – 3x + 18. Hence, find all possible values of x for which f(x) = 0.
Solution
f(x) = 2x3 – 7x2 – 3x + 18
For x = 2,
f(x) = f(2)
= 2(2)3 – 7(2)2 – 3(2) + 18
= 16 – 28 – 6 + 18
= 0
Hence, (x – 2) is a factor of f(x).
2x2 – 3x – 9
`x - 2")"overline(2x^3 - 7x^2 - 3x + 18)`
2x3 – 4x2
– +
– 3x2 – 3x
– 3x2 + 6x
+ –
– 9x + 18
– 9x + 18
+ –
0
∴ 2x3 – 7x2 – 3x + 18 = (x – 2)(2x2 – 3x – 9)
= (x – 2)(2x2 – 6x + 3x – 9)
= (x – 2)[2x(x – 3) + 3(x – 3)]
= (x – 2)(x – 3)(2x + 3)
Now, f(x) = 0
⇒ 2x3 – 7x2 – 3x + 18 = 0
⇒ (x – 2)(x – 3)(2x + 3) = 0
⇒ `x = 2, 3, (-3)/2`
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