Polynomials

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					POLYNOMIALS
I. Exponents
  A. Definitions

     Exponents are shorthand notation for repeated multiplication
             a   
      a n  a   a  a .
             n factors
     Zero exponent b0  1
                                  1
     Negative exponent b  n 
                                  bn

  To see why the zero exponent and negative exponents are defined this way
  consider the pattern in the example below. In the left column the exponents
  decrease by 1 and in the right column the numbers decrease by a factor of 10.

                                  103  1000
                                  10 2  100
                                  101  10
                                  10 0  1
                                              1    1
                                  10 1          1
                                             10 10
                                               1     1
                                  10  2          2
                                             100 10
                                                1     1
                                  10 3            3
                                             1000 10

  B. Rules of Exponents

       Product rule (same base)                a ma n  a mn
                                               am       m n
                                                  n  a
       Quotient rule (same base)
                                               a
       Product rule (same exponent)            ( ab) m  a m b m
                                                    m
                                                a am
       Quotient rule (same exponent)              m
                                                b b
       Power rule                              a 
                                                 m n
                                                         a mn
II. Polynomials
   A polynomial has terms consisting of constants and variables raised to
   positive integer powers. Polynomials are classified according to the
   number of terms they have. Monomials have one term, binomials have two
   terms and trinomials have three terms. The degree of a polynomial in one
   variable is the greatest exponent. The degree of a polynomial of several
   variables is the greatest sum of the exponents that occurs in any of the
   terms.

III. Addition/Subtraction
   Like terms have the same variables with the same exponents.
   Polynomials are added/subtracted by combining like terms.

IV. Multiplication
  A. Multiply a polynomial by a monomial

   Multiply each term of the polynomial by the monomial. This is the same as
   saying use the distributive rule.

  B. Multiply a polynomial by a polynomial

   FOIL Method.
   When multiplying binomials use first, outer, inner and last and combine
   like terms.

   General Method.
   Multiply each term of the second polynomial by each term of the first
   polynomial and combine like terms.

V. Factoring
   Choose a factoring technique based on the number of terms.

  A. Greatest common factor

   Always factor out the GCF before moving on to other techniques. Factor
   out the largest number that divides into each term evenly. Factor out the
   variables that are common to all the terms using the lowest exponent.
B. Factoring formulas (two terms)

    Difference of squares         x 2  a 2  ( x  a )( x  a )
    Difference of cubes           x 3  a 3  ( x  a )( x 2  ax  a 2 )
    Sum of cubes                  x 3  a 3  ( x  a )( x 2  ax  a 2 )

C. Factoring trinomials (three terms)

   When the polynomial has three terms use the trial and error method.

   Leading coefficient is a 1.

   Since x 2  (a  b) x  ab  ( x  a)( x  b) to factor x 2  bx  c we must find
   numbers whose sum is b and whose product is c. In other words find two
   numbers that multiply to the last term and add to give the coefficient of
   the middle term.

   Leading coefficient is not a 1.

      Firsts must multiply to give the first term.
      Lasts must multiply to give the third term.
      The inner and outer must sum to give the second term.

   Comments:
   (1) The AC test states that ax 2  bx  c is factorable only if there are two
   numbers whose product is ac and whose sum is b. For example
   3x 2  2 x  5 is not factorable since there are no numbers, which multiply
   to 15 and add to 2.
   (2) The slide technique as shown in the example below has less trial and
   error for the case in which the leading coefficient is not 1.

             6 x 2  x  12         Multiply the last term by the
             x 2  x  72           leading coefficient.
                                    Factor and then divide both
                  9      8      numbers      by    the leading
              x   x  
                  6      6      coefficient.
                  3      4      Reduce the fractions.
              x   x           Multiple the denominators up
                  2      3
             (2 x  3)(3 x  4)     to the variables.
  D. Factoring by grouping (four or more terms)

   Rearrange the terms (if necessary) so that the first two terms have a
   common factor and the last two terms have a common factor. Factor out the
   GCF from the first two terms and the second two terms. Factor a binomial
   factor from the remaining terms.

VI. Solving Equations

   Use the zero factor property. If ab  0 then a  0 or b  0 .

VII. Division
  A. Division by a monomial

    Divide each term by the monomial

  B. Division by a polynomial

    Use the long division algorithm.

  C. Synthetic division

    Used to divide a polynomial by a binomial of the form x  c .

    Ex. Divide 2x 4  3x 2  5x  7 by x  3 .

    3 2     0    3      5 7
          6      18  45 120
        2 6       15  40 113

                                 113
    2 x 3  6 x 2  15x  40 
                                 x3

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