Section 4.3 - Zeros of Polynomials

Section Objectives

  1. Apply the Fundamental Theorem of Algebra.
  2. Use common tests to determine the nature of a polynomial's zeros.
  3. Factor a polynomial with real coefficients into a product of linear factors and irreducible quadratic factors.



Rational Zeros Theorem

If is a polynomial with integer coefficients, then any rational zero of must have the form where is a factor of and is a factor of .



Example




To find the rational zeros of a polynomial with integer coefficients:

  1. List all rational zeros.
  2. If appropriate, use a graphing calculator to narrow done the list.
  3. When very simple to do so, check a possible zero by directly evaluating the polynomial.
  4. When synthetic division is more appropriate, check a possible zero by using synthetic division to evaluate the polynomial.



Example




Some Other Useful Tests

Descartes' Rule of Signs: When a polynomial with real coefficients is written is order of decreasing exponents,



Examples







Intermediate Value Theorem: If is a polynomial and and are two real numbers for which and have opposite signs, then has at least one real zero between and .



Example




Fundamental Theorem of Algebra

If is a polynomial of degree , , then has at least one zero.


Some important results follow from the fundamental theorem:



Conjugate Roots Theorem: Suppose is a polynomial with real coefficients. If the complex number is a zero of , then the complex number is also a zero of . Thus the complex zeros of a polynomial with real coefficients always occur in conjugate pairs.



A quadratic factor of a polynomial with real coefficients is said to be irreducible if its zeros are imaginary complex conjugates.



Examples







Factoring Polynomials

Finding all real and complex zeros of a polynomial is typically a difficult task. Here are some guidelines:

  1. Start by factoring as much as is easily possible.
  2. Use the quadratic formula if applicable.
  3. Apply the Rational Zeros Theorem (or other tests).
  4. Test the possible rational zeros and, each time you find one, use synthetic division to deflate the polynomial (divide out the corresponding linear factor).
  5. Repeat steps 1-4, as necessary, on the deflated polynomial.
  6. Use a graphing calculator to help locate the real zeros.
  7. If necessary, apply more advanced zero-finding techniques.



Examples