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Question
Using the Remainder Theorem, factorise the expression 3x3 + 10x2 + x – 6. Hence, solve the equation 3x3 + 10x2 + x – 6 = 0
Solution
Let f(x) = 3x3 + 10x2 + x – 6
For x = –1,
f(x) = f(–1)
= 3(–1)3 + 10(–1)2 + (–1) – 6
= –3 + 10 – 1 – 6
= 0
Hence, (x + 1) is a factor of f(x).
3x2 + 7x – 6
`x + 1")"overline(3x^3 + 10x^2 + x - 6)`
3x3 + 3x2
– –
7x2 + x
7x2 + 7x
– –
– 6x – 6
– 6x – 6
+ +
0
∴ 3x3 + 10x2 + x – 6 = (x + 1)(3x2 + 7x – 6)
= (x + 1)(3x2 + 9x – 2x – 6)
= (x + 1)[3x(x + 3) – 2(x + 3)]
= (x + 1)(x + 3)(3x – 2)
Now, 3x3 + 10x2 + x – 6 = 0
(x + 1)(x + 3)(3x – 2) = 0
`x = -1, -3, 2/3`
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