COLLEGE ALGEBRA
MATH 110
Southwestern   
Adventist University 
 
   Distance Education Lawrence E. Turner, Jr., Ph.D.  


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Zeros of Polynomials


A polynomial of degree n (n is a positive integer):

   f(x)  = anxn + an−1xn−1 +  ... +  a1x + a0,       an ≠ 0.
  1. has exactly n zeros (not all may be distinct)
  2. any complex zeros must occur in pairs, hence there are an even number of complex zeros
  3. if n is odd, then there must be at least one real zero
Factoring the polynomial will result in:

  f(x) = an(x−r1)(x−r2) ... (x−rk)2 ... (x2+bkx+ck) ...

where r1 and r2 are examples of distinct real zeros, rk is an example of a zero that occurs with a multiplicity of two, and the irreducible quadratic factor results in two complex zeros occurring as a complex conjugate pair.

To determine zeros:
  1. Try to factor out any common factors—both numbers from the coefficients and any x's.
    eg. if there is an x in every term, then x = 0 is a zero.
  2. From Descartes' Rule of Signs on f(x), get possible numbers of positive real zeros.
  3. From Descartes' Rule of Signs on f(−x), get possible number of negative real zeros.
  4. Construct a table of possible configurations of zeros.
  5. Evaluate the polynomial at 0, this is easy: f(0) = a0.
  6. Evaluate the polynomial at positive integer values: 1, 2, 3, .... Use synthetic division. You can stop when you find an positive integer such that all the numbers in the quotient row including the remainder are nonnegative—this is an upper bound, (UB). If you get a remainder of zero, then you have found a zero of the polynomial—go to step 11.
  7. Evaluate the polynomial at negative integer values: −1, −2, −3, .... Use synthetic division. You can stop when you find a negative integer such that all the numbers in the quotient row including the remainder alternate in sign—this is a lower bound, (LB). If you get a remainder of zero, then you have found a zero of the polynomial—go to step 11.
  8. Construct a table of all possible rational zeros of the form b/c where b is a factor of a0 (the coefficient of the constant term) and c is a positive factor of an (the coefficient of the leading term). Eliminate redundant possible rational zeros.
  9. Eliminate potential rational zeros by using the upper bound (i.e those that are greater) from step 6 and the lower bound (i.e. those that are smaller) from step 7.
  10. Use the intervals where the function changes sign, from steps 5, 6, and 7, to determine where potential zeros are. Test all the potential rational zeros from step 8 that are in the potential interval(s). If you get a remainder of zero, then you have found a zero of the polynomial—go to step 11.

  11. Whenever you find a zero, x = r, then (x−r) is a factor of the polynomial. Factor it out (using synthetic division), and reduce the degree of the resulting polynomial by one. The other factor is the quotient with degree n−1, and its coefficients are in the quotient row. Then start over with the simpler polynomial—go to step 2, although you have done much of the work and probably can easily modify the results of that step and the following ones by simply accounting for the zero you have found.

  12. If you end up with a quadratic, then you can use the quadratic equation to find its two zeros, which may be a complex conjugate pair (or may be two real values, rational or irrational).
 
© 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by Lawrence Turner