Final: The Full Sweep

To factor an expression completely, work in layers. First factor out the greatest common factor of all the terms. Then factor the quadratic that is left, using the signs of its middle term and its constant to choose the factor pair. Finally, check whether any factor can be factored again.

For example, every term of x^3 + 2x^2 - 15x has a factor of x, so start with x(x^2 + 2x - 15). The constant -15 is negative, so the two numbers have opposite signs. The pair 5 and -3 multiplies to -15 and adds to 2. The complete factoring is x(x + 5)(x - 3). Neither linear factor can be factored again, so the factoring is finished.