Learning Resources

Focus D: Solving Polynomial Equations by Factoring

A few points to remember about synthetic substitution:

• synthetic substitution may have to be done twice. The usual goal is to get down to a quadratic equation, the roots to which you can easily find by factoring or the quadratic formula. If you begin with a quartic polynomial, doing synthetic substitution once will result in a cubic. Repeating the process will result in a quadratic.
• According to the Factor Theorem, if the numerical value by which you are dividing is indeed a root, then the remainder will always be zero. If you are certain that you have identified a root, yet synthetic substitution doesn't produce  a remainder of zero, check the process for errors!
• If the polynomial is missing a term, you must supply a zero for its coefficient.
• Once you have found the first factor of a polynomial, synthetic substitution is an easy process by which to find the remaining factors.
• If you find this method confusing initially, keep practicing the process. With practice, it will become the lest complicated method and, without a doubt, your choice of preference.