Factorization of Cubic Equations

a3+b3=(a+b)(a2-ab+b2) a3-b3=(a-b)(a2+ab+b2) Solving a cubic Write f(x)=0. Use a given root, or test candidates with the factor theorem. If the constant term is zero, factor out x. For integer coefficients, a rational root pq in lowest terms has p dividing the constant and q dividing the leading coefficient. A cubic need not have a rational root. If f(k)=0, divide by x-k, then solve the quadratic factor. **Quadratic factor** | **Working** | **Roots of the cubic** Factorises | (x-1)(x2-x-6)=0 (x-1)(x-3)(x+2)=0 | x=1, x=3, x=-2 Does not factorise using integers, b2-4ac>0 | (x+2)(x2-2x-1)=0 x=-2 or x= 2+-sqrt(8)2 | x=-2, x=1+-sqrt(2) b2-4ac<0 | (x-2)(x2+x+5)=0 1-4(1)(5)=-19<0 | x=2 only Over the reals, leave a quadratic factor with D<0 unfactorised.