The Rational Zero Theorem gives you a complete list of candidates for rational zeros of P(x)

When can I use it? P(x) must have Integer Coefficients only – no irrational coefficients

With , where is he leading coefficient and is the trailing coefficient, and all are integers, no fractions.

The polynomial must have only integer coefficients. If it has any rational number coefficients, then multiply each term by the LCD of the fraction coefficients. The new polynomial will have the same zeros and the Rational Zero Theorem may be used on it.

How do I use it?

  • Find all the factors of the trailing coefficient, . And find all the factors of the leading coefficient, .
  • The candidates are all the possible fractions formed by , positive and negative both.

The list of candidates is often written in shorthand like this: . (See specific example, below.)

When is it impractical to use?

If there are a lot of factors, the number of possible zeros is huge. Example: .
The list of combinations , you get . There are 128 candidates, counting duplicates.

Other tools can help save you some work

061_RationalZeroTheorem.docx 12/23/2013 1:48 PM D.R.S.

  • Descartes’s Rule of Signs. It could tell you that all the real number zeros are positive, or maybe all the real number zeros are negative. That would cut the size of the list in half. Or if it indicates exactly one positive real zero or one negative real zero, you need just find that one and then the rest of the candidates with that sign are eliminated.
  • The Bounded Zero Theorem might be able to constrain the low-to-high range of candidates that need to be checked.
  • The TI-84 calculator graphing and table of values can help.
  • Good guessing can help you find zeros more quickly. Be aware that smaller numbers like 1, 2, 3, 4, 5, and fractions made of smaller numbers, occur more often in most problems. So you test earlier and postpone testing a candidate like until later.
  • When you do find one of the zeros, divide out the factor from to obtain a new polynomial that’s one degree lower. The numbers may be smaller and the resulting candidate list may be shorter.
  • When you do find one of the zeros, test that same zero again on the new polynomial. It could well be a zero of multiplicity two or higher.
  • Limitation: this only finds rational number zeros. It won’t help in finding irrational or complex zeros.

061_RationalZeroTheorem.docx 12/23/2013 1:48 PM D.R.S.