Section 2-4 by v7166R

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									   Section 2-4
Real Zeros of Polynomial
       Functions
             Section 2-4
• long division and the division algorithm
• the remainder and factor theorems
• reviewing the fundamental connection for
     polynomial functions
• synthetic division
• rational zeros theorem
• upper and lower bounds
             Long Division
• long division for polynomials is just like
  long division for numbers
• it involves a dividend divided by a divisor
  to obtain a quotient and a remainder
• the dividend is the numerator of a fraction
  and the divisor is the denominator
           Division Algorithm
• if f(x) is the dividend, d(x) is the divisor,
  q(x) the quotient, and r(x) the remainder,
  then the division algorithm can be stated
  two ways
               f ( x)  d ( x)  q( x)  r ( x)
                             or
               f ( x)           r ( x)
                       q( x) 
               d ( x)           d ( x)
         Remainder Theorem
• if a polynomial f (x) is divided by x – k, then
  the remainder is f (k)
• in other words, the remainder of the division
  problem would be the same value as plugging
  in k into the f (x)
• we can find the remainders without having to
  do long division
• later, we will find f (k) values without having
  to plug k into the function using a shortcut for
  long division
            Factor Theorem
• the useful aspect of the remainder theorem is
  what happens when the remainder is 0
• since the remainder is 0, f (k) = 0 which
  means that k is a zero of the polynomial
• it also means that x – k is a factor of the
  polynomial
• if we could find out what values yield
  remainders of 0 then we can find factors of
  polynomials of higher degree
         Fundamental Connection
    For a real number k and a polynomial function
    f (x), the following statements are equivalent
•    k is a solution (or root) of the equation f (x) = 0
•    k is a zero of the function f (x)
•    k is an x-intercept of the graph of f (x)
•    x – k is a factor of f (x)
           Synthetic Division
• finding zeros and factors of polynomials
  would be simple if we had some easy way
  to find out which values would produce a
  remainder of 0 (long division takes too
  long)
• synthetic division is just that shortcut
• it allows us to quickly divide a function
  f (x) by a divisor x – k to see if it yields a
  remainder of 0
          Synthetic Division
• it follows the same steps as long division
  without having to write out the variables
  and other notation
• it is really fast and easy
• if a zero is found, the resulting quotient is
  also a factor, and it is called the depressed
  equation because it will be one degree
  less than the original function
        Synthetic Division
Divide 2 x  3x  5x  12 by x  3
          3    2
         Synthetic Division
Divide 2 x  3x  5x  12 by x  3
          3    2


3    2   -3   -5    - 12
         Synthetic Division
Divide 2 x  3x  5x  12 by x  3
          3    2


3    2   -3   -5    - 12

          6     9    12

    2     3    4      0
            Synthetic Division
Divide 2 x  3x  5x  12 by x  3
              3       2


3     2     -3       -5     - 12

              6       9       12

      2       3       4        0

The remainder is 0 so x – 3 is a factor and
the quotient, 2x2 + 3x + 4, is also a factor
    Rational Zeros Theorems
• if you want to find zeros, you need to
  have an idea about which values to test in
  S.D. (synthetic division)
• the rational zeros theorem provides a list
  of possible rational zeros to test in S.D.
• they will be a value  p        where,
                           q
     p must be a factor of the constant
     q must be a factor of the leading term
Finding Possible Rational Zeros
 Find all the possible rational zeros of
  f ( x)  3x  4 x  5 x  2
            3     2



  p  2 and q  3
  factors of p: 1, 2
  factors of q: 1, 3
                                   1 2
  possible rational zeros:  1, 2, ,
                                   3 3
     Upper and Lower Bounds
• a number k is an upper bound if there are
  no zeros greater than k; if k is plugged into
  S.D., the bottom line will have no sign
  changes
• a number k is a lower bound if there are
  no zeros less than k; if k is plugged into
  S.D., the bottom line will have alternating
  signs (0 can be considered + or -)
    Upper and Lower Bounds
• if you are looking for zeros and you come
  across a lower bound, do not try any
  numbers less than that number
• if you are trying to find zeros and you
  come across an upper bound, do not try
  any numbers greater than that number

								
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