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CHAPTER 3_ POLYNOMIAL AND RATIONAL FUNCTIONS

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					           CHAPTER 3: POLYNOMIAL AND RATIONAL FUNCTIONS

3.1       POLYNOMIAL FUNCTIONS AND MODELING

         Polynomial Function
          A polynomial function P is given by
                 P( x)  an xn  an1xn1  an2 xn2     a1x  a0 ,
          where the coefficients an , an 1 ,..., a1 , a0 are real numbers and the
          exponents are whole numbers.

              o The first nonzero coefficient, an , is called the leading
                coefficient
              o The term an xn is called the leading term
            o The degree of the polynomial function is n
         The Leading Term Test
            o The behavior of the graph of a polynomial function as x
                becomes very large ( x   ) or very small ( x   ) is referred
                to as the end behavior of the graph.
                    The leading term determines a graph’s end behavior
            o The Leading Term Test
                  If an xn is the leading term of a polynomial function, then the
                  behavior of the graph as x   or x   can be described in
                  one of the four following ways.


                          1.      If n is even, and an  0 :




                          2.      If n is even, and an  0 :
                          3.    If n is odd, and an  0 :




                          4.    If n is odd, and an  0 :




         Even and Odd Multiplicity
          If  x  c  , k  1, is a factor of a polynomial function P( x) and  x  c 
                      k                                                                    k 1


          is not a factor and
              o k is odd, then the graph crosses the x-axis at  c, 0  ;
             o k is even, then the graph is tangent to the x-axis at  c, 0  .
         Polynomial Models
             o Cubic Regression
             o Quartic Regression

3.2       GRAPHING POLYNOMIAL FUNCTIONS

         Graphing Polynomial Functions
          If P( x) is a polynomial function of degree n, the graph of the function
          has at most n real zeros, and therefore at most n x-intercepts; at
          most n  1 turning points.

             o Steps for Graphing Polynomial Functions
                 1.       Use the leading-term test to determine the end behavior.
                 2.       Find the zeros of the function by solving f ( x)  0. Any
                          real zeros are the first coordinates of the x-intercepts.
                 3.       Use the zeros (x-intercepts) to divide the x-axis into
                          intervals and choose a test point in each interval to
                          determine the sign of all function values in that interval.
                  4.       Find f (0). This gives the y-intercept of the function.
                  5.       If necessary, find additional function values to determine
                           the general shape of the graph and then draw the graph.
                  6.       As a partial check, use the facts that the graph has at
                           most n x-intercepts and at most n  1 turning points.
                           Multiplicity of zeros can also be considered in order to
                           check where the graph crosses or is tangent to the x-
                           axis. We can also check the graph with a graphing
                           calculator.



         The Intermediate Value Theorem
          For any polynomial function P( x) with real coefficients, suppose that
          for a  b , P (a ) and P(b) are of opposite signs. Then the function has
          a real zero between a and b.



3.3       POLYNOMIAL DIVISION; THE REMAINDER AND FACTOR
          THEOREMS

         This section teaches us concepts that help to find the exact zeros of
          polynomial functions with degree three or higher.

         Consider the function

          h( x)  x3  2 x 2  5 x  6   x  3 x  1 x  2 


             This gives us the following zeros:

               x  3  0  x  3
               x  1  0  x  1           3, 1, 2
               x  2  0  x  2

             When you divide one polynomial by another, you obtain a quotient
              and a remainder.
        o If the remainder is zero then the divisor is a factor of the
          dividend.

              P ( x)  d ( x)  Q ( x)  R ( x)
              P ( x) : Dividend
              d ( x) : Divisor
              Q ( x) : Quotient
              R ( x) : Remainder
    Synthetic Division
        o Consider the following:

               4x   3
                          3x 2  x  7    x  2  .
        A.
                           4 x 2  5 x  11
              x  2 4 x3  3x 2              x 7
                          4 x3  8 x 2
                                  5x 2       x
                                  5x  10 x
                                     2


                                           11x  7
                                           11x  22
                                                  29

        B.
                          4 5 11
              1 2 4  3  1 7
                         48
                           5 1
                             5  10
                                 11  7
                                  11  22
                                         29



C.
                     2      4       3            1       7
                                      8       10          22
                            4         5      11           29


         The Remainder Theorem

          If a number c is substituted for x in the polynomial f ( x) , then the
          result f (c) is the remainder that would be obtained by dividing
          f ( x) by x  c. In other words, if f ( x)  ( x  c)  Q( x)  R, then
          f (c)  R.


         The Factor Theorem

          For a polynomial f ( x) , if f (c)  0, then x  c is a factor of f ( x).


             Proof: If we divide f ( x) by x  c, we obtain a quotient and a
             remainder, related as follows:

                     f ( x)  ( x  c)  Q( x)  f (c).

             Then if f (c)  0, we have

                     f ( x)  ( x  c)  Q( x),

             so x  c is a factor of f ( x) .



3.4   THEOREMS ABOUT ZEROS OF POLYNOMIAL FUNCTIONS

         The Fundamental Theorem of Algebra
          Every polynomial function of degree n, with n  1, has at least one
          zero in the system of complex numbers.


         Every polynomial function f of degree n, with n  1, can be factored
          into n linear factors (not necessarily unique); that is,
           f ( x)  an  x  c1  x  c2   x  cn  .
       Finding Polynomials with Given Zeros

           o If a complex number a  bi, b  0, is a zero of a polynomial
               function f ( x) with real coefficients, then its conjugate,
               a  bi , is also a zero.
                    Example: Find a polynomial function of degree 3,
                        having the zeros 1, 3i , and 3i.

                       Solution: f ( x)  an  x  1 x  3i  x  3i  .
                                    The number an can be any nonzero number.
                                    The simplest function will be obtained if we
                                    let an  1. Then we have
                                     f ( x)   x  1 x  3i  x  3i 
                                             x  1  x 2  9 
                                            x3  9 x  x 2  9
                                            x3  x 2  9 x  9

   Rational Coefficients

    If a  b c , where a and b are rational and c is not a perfect square, is
    a zero of a polynomial function f ( x) with rational coefficients, then
    a  b c is also a zero.

   Integer Coefficients and the Rational Zeros Theorem
       o The Rational Zeros Theorem
           Let P( x)  an xn  an1x n1  an2 xn2        a1x  a0 , where all the
           coefficients are integers. Consider a rational number denoted
           by p , where p and q are relatively prime (having no common
                q
           factor besides 1 and -1). If p is a zero of P( x) , then p is a
                                         q
           factor of a0 and q is a factor of an .
                          Example: Given f ( x)  3x 4  11x3  10 x  4 :
                             a) Find the rational zeros and then the other
                                zeros; that is, solve f ( x)  0.
                     b) Factor f ( x) into linear factors.

                     Solution:
                     a) Because the degree of f ( x) is 4, there are
                             at most 4 distinct zeros. The possibilities
                             for p are
                                    q
possibilities for p    1, 2, 4
                    :             ;
possibilities for q     1, 3
                                             1   1 2     2 4     4
possibilities for p : 1,  1, 2,  2, 4,  4, ,  , ,  , , 
                     q                       3   3 3     3 3     3

                          Now we need to graph the function on a
                          graphing calculator to see which of these
                          possibilities seem to be zeros.




                          When you divide the function by
                          x  (1) or x  1 , you’ll find that -1 is a zero.
                          Dividing the quotient 3x3  14 x 2  14 x  4
                          (obtained from the first division problem
                                            1                   1
                          above) by x         will show that     is not a
                                            3                   3
                                                          2                2
                          zero. Repeating again for          shows that
                                                          3                3
                          is a zero. We obtain a quotient of
                                                                2
                          3x 2  12 x  6 . So we have 1 and      as the
                                                                3
                          rational zeros and  3x 2  12 x  6  factors as
                           3  x 2  4 x  2  , which yields 2  2 as the
                          other two zeros.
                     b)   Therefore the complete factorization of
                           f ( x) is
                                                               2
                                        f ( x)   x  1  x    3x 2  12 x  6 
                                                               3
                                                               2
                                                 x  1  x    3  x 2  4 x  2 
                                                               3
                                                            
                                                            
                                                                 2
                                                                                          
                                                3  x  1  x    x  2  2   x  2  2 
                                                                 3
                                                                                           



3.5       RATIONAL FUNCTIONS

         A rational function is a function f that is a quotient of two
                                         p( x)
          polynomials, that is, f ( x)        , where p  x  and q  x  are polynomials
                                         q( x)
          and where q  x  is not the zero polynomial. The domain of f consists
          of all inputs x for which q( x)  0.


         Asymptotes

             o Vertical Asymptotes
                   The line x  a is a vertical asymptote for the graph of f
                        if any of the following are true:

                         f ( x)   as x  a  or f ( x)   as x  a  , or
                         f ( x)   as x  a  or f ( x)   as x  a  .

                           Determining Vertical Asymptotes
                                                             p( x)
                            For a rational function f ( x)        , where p( x) and
                                                             q( x)
                            q ( x ) are polynomials with no common factors other
                            than constants, if a is a zero of the denominator, then
                            the line x  a is a vertical asymptote for the graph of
                            the function.



                Horizontal Asymptotes
                         Determining a Horizontal Asymptote
                               When the numerator and the denominator of
                                a rational function have the same degree,
                                             a
                                the line y  is the horizontal asymptote,
                                             b
                                where a and b are the leading coefficients
                                of the numerator and the denominator,
                                respectively.
                               When the degree of the numerator of a
                                rational function is less than the degree of
                                the denominator, the x-axis, or y  0, is the
                                horizontal asymptote.
                               When the degree of the numerator of a
                                rational function is greater than the degree
                                of the denominator, there is no horizontal
                                asymptote.
           Oblique, or Slant, Asymptotes
                        Example: Find all asymptotes of

                                                   2 x 2  3x  1
                                        f ( x) 
                                                        x2

                         Solution:
                            1. Since x  2  0  x  2 , this gives us a vertical
                               asymptote at the line x  2.
                            2. There are no horizontal asymptotes since
                               the degree of the numerator is greater than
                               the degree of the denominator.
                            3. Dividing: (2 x 2  3x  1)  ( x  2) , we get
                                              1
                                 2 x  1       . Now we see that when
                                             x2
                                                       1
                                x   or x  ,            0 and the value of
                                                      x2
                                 f ( x)  2 x  1. This means that as the
                                absolute value of x becomes very large, the
                                graph of f ( x) gets very close to the graph
                                of y  2 x  1. Thus the line y  2 x  1 is the
                                oblique asymptote.
3.6   POLYNOMIAL AND RATIONAL INEQUALITIES
   Polynomial Inequalities
       o To solve a polynomial inequality:
             1. Find an equivalent inequality with 0 on one side.
             2. Solve the related polynomial equation.
             3. Use the solutions to divide the x-axis into intervals.
                 Then select a test value from each interval and
                 determine the polynomial’s sign on the interval.
             4. Determine the intervals for which the inequality is
                 satisfied and write interval notation or set-builder
                 notation for the solution set. Include the endpoints of
                 the intervals in the solution set if the inequality symbol is
                  or  .

   Rational Inequalities

       o To solve a rational inequality:

              1. Find an equivalent inequality with 0 on one side.
              2. Change the inequality symbol to an equals sign and solve
                 the related equation.
              3. Find the values of the variable for which the related
                 rational function is not defined.
              4. The numbers found in steps (2) and (3) are called critical
                 values. Use the critical values to divide the x-axis into
                 intervals. Then test an x-value from each interval to
                 determine the function’s sign in that interval.
              5. Select the intervals for which the inequality is satisfied
                 and write interval notation or set-builder notation for
                 the solution set. If the inequality symbol is  or , then
                 the solutions from step (2) should be included in the
                 solution set. The x-values found in step (3) are never
                 included in the solution set.
3.7       VARIATION AND APPLICATIONS

         Direct Variation
          If a situation gives rise to a linear function f ( x)  kx or y  kx, where k
          is a positive constant, we say that we have direct variation, or that y
          varies directly as x, or that y is directly proportional to x. The
          number k is called the variation constant, or constant of
          proportionality.

         Inverse Variation
                                                             k        k
                                                                or y  , where k
          If a situation gives rise to a linear function f ( x) 
                                                              x       x
          is a positive constant, we say that we have inverse variation, or that y
          varies inversely as x, or that y is inversely proportional to x. The
          number k is called the variation constant, or constant of
          proportionality.

         Combined Variation

          y varies directly as the nth power of x if there is some positive
          constant k such that y  kx n .

          y varies inversely as the nth power of x if there is some positive
                                    k
          constant k such that y  n .
                                   x

          y varies jointly as x and z if there is some positive constant k such
          that y  kxz.

				
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