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Question
Using the Remainder Theorem, factorise the following completely:
3x3 + 2x2 – 23x – 30
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Solution
f(x) = 3x3 + 2x2 – 23x – 30
For x = –2,
f(x) = f(–2)
= 3(–2)3 + 2(–2)2 – 23(–2) – 30
= –24 + 8 + 46 – 30
= –54 + 54
= 0
Hence, (x + 2) is a factor of f(x).
3x2 – 4x – 15
`x + 2")"overline(3x^3 + 2x^2 - 23x - 30)`
3x3 + 6x2
– 4x2 – 23x
– 4x2 – 8x
– 15x – 30
– 15x – 30
0
∴ 3x3 + 2x2 – 23x – 30 = (x + 2)(3x2 – 4x – 15)
= (x + 2)(3x2 + 5x – 9x – 15)
= (x + 2)[x(3x + 5) – 3(3x + 5)]
= (x + 2)(3x + 5)(x – 3)
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