x^3 + 2x^2 - 9x - 18 = (x^2 - a^2)(x + b) where a,b are integers. Work out the three linear factors of x^3 + 2x^2 - 9x - 18. (Note: x^3 indicates x cubed and x^2 indicates x squared).

There are a few different ways to approach this problem. The most obvious is to attempt to factorise x+ 2x- 9x - 18. However it is very difficult to approach the problem like this. fortunately the question has given us that the cubic expression factorises to(x2-a2)(x+b). If we expand this back out we get x+ bx- a2x - a2b. We can then compare this cubic to our original and see that a2 = 9 and b = 2.

So we now have x3+2x2-9x-18 = (x2-9)(x+2). We know that we can factorise x2-9 to (x+3)(x-3) so our linear factorisation of the original cubic is (x+3)(x-3)(x+2).

CB
Answered by Chris B. Further Mathematics tutor

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