Real roots of cubic equations

Solve P(x)=0 by writing P as a linear factor times a quadratic and using the zero-product rule. Do not divide by a factor containing x: that may discard a root. Why: a product of real numbers is zero exactly when at least one factor is zero. For the quadratic cofactor, positive discriminant gives two distinct real roots, zero gives a repeated root, and negative gives no real roots. Method: put every term on one side; find/divide out a linear factor; solve the quadratic by factorisation or the quadratic formula; list each distinct real solution once and check whether roots from different factors coincide. Example: x3-3x+2=(x-1)(x2+x-2)=(x-1)2(x+2). Hence the distinct real roots are 1 and -2; 1 is repeated. In contrast, (x-1)(x2+1)=0 has only the real root 1, since the quadratic discriminant is -4. A cubic need not have three distinct real solutions. Check: keep exact surds until approximation is requested; a repeated root is not an extra distinct solution.