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Precalculus: Roots of Polynomials

Click on title for full document summarizing all concepts. 

(From Mr. James Jones, Professor of Mathematics at Richland Community College, Decatur, IL)

 

Suggested Attack to Finding Zeros of a Polynomial

1. Identify the total number of real or complex zeros (corollary to Fundamental Theorem of Algebra).
2. Identify the possible number of positive, negative, and complex zeros (Descartes' Rule of Signs).
3. List the possible rational zeros (Rational Root Theorem)
4. Try possible rational zeros until you find one that works. After each division by a positive value, check for possible upper bounds. After each division by a negative value, check for possible lower bounds (Upper and Lower Bound Theorems)
5. After you find a possible rational root that actually works, take the quotient and continue to try to factor it until it is down to a quadratic or less. Once it is a quadratic or less, there are other ways to solve it.
6. Write the linear and or linear / irreducible quadratic factorization (next section)

Really Important (and frustrating if you forget)!

Once you have found a zero using synthetic division, use the quotient as a new polynomial for all further divisions. The quotient will be one less degree than the original dividend. Each time you find a root, the quotient becomes one less in degree. Eventually, it will become a quadratic, and then you can factor, extract roots, complete the square, or use the quadratic equation to find the remaining roots.

If you continue to use the original function, you will become very frustrated and waste a lot of time.