factorise x^3 + 3x^2 - 13x - 15
For convience we can call the polynomial f(x). Using the fact that the product of the roots of a polynomial equals the constant term, we know that the product of the roots is -15. We can therefore guess a root of the polynomial by considering the value of f(x) when x is a factor of -15, ie. x = 1, 3, 5, 15, -1, -3, -5, -15. f(3) = 27 + 27 - 39 -15 = 0 so we know that 3 is a root of f(x) and so we can deduce that (x-3) is a factor of f(x). Therefore we can write f(x) = (x-3)(x2 + Ax + 5) where A is unknown. To find A we can work out what the coefficient of the x term would be if we expanded the above: 5 + (-3)A = 5 - 3A and comparing this with the x coefficient in the original form of f(x), we know 5 - 3A = -13 and so A = 6. Therefore we have f(x) = (x-3)(x2 + 6x + 5) which we know how to factorise, giving us f(x) = (x-3)(x+5)(x+1)
