Complete cubic factorisation

For a cubic, first find a linear factor and its quadratic cofactor. Factor the quadratic further whenever it has real linear factors. Why: if P(r)=0, the division identity becomes P(x)=(x-r)Q(x), with Q quadratic. Multiplying the factors recovers the original cubic. Method: first check for a common factor or grouping; otherwise test a given or simple rational root and divide. Factor the quadratic, or use its discriminant to decide whether it stays irreducible over the reals. Example: x3+2x2-x-2=x2(x+2)-(x+2)=(x+2)(x2-1)=(x+2)(x-1)(x+1). Useful identities: u3-v3=(u-v)(u2+uv+v2), u3+v3=(u+v)(u2-uv+v2). Expand to check the middle terms cancel. Check: stopping at a quadratic such as x2-1 is incomplete. A quadratic with negative discriminant has no real linear factors; leave it intact.