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3.2 Polynomials-- Properties of Division

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					3.2 Polynomials-- Properties of
          Division
   Leading to Synthetic Division
                Part 1
Some serious relationships
               Start with a POD
Using long division,
       divide x4 - 16 by x2 + 3x + 1



By hand.

On CAS.
               Start with a POD
Using long division,
       divide x4 - 16 by x2 + 3x + 1

Quotient: x2 - 3x + 8

Remainder: -21x - 24
              The POD leads to
The Division Algorithm for Polynomials:

      f(x) = d(x) · q(x) + r(x)

      where d, q, and r are polynomials,

      d is the divisor, q is the quotient, r is the
      remainder,

      and r(x) has a smaller degree than d(x).
      There’s a pattern– what is it?
Find the remainders with LINEAR divisors:


         x3  2 x 2  4 x  8      x  3 x3  2 x 2  4 x  8
                x 3
                     Compare that to f(3)– the top polynomial.


     2 x 4  3x 3  x 2  x  9
                                   x  4 2 x 4  3x 3  x 2  x  9
               x4
                     Compare that to f(-4).
                  The Relationship
The Division Algorithm for Polynomials:

        f(x) = d(x) · q(x) + r(x)

        What happens when the remainder = 0?

The Factor Theorem:

        If the remainder = 0, then f(c) = 0.

        If f(c) = 0, then c is a root/ solution/ zero of f(x) and (x - c)
        is a factor of f(x).
                                 The Proof
Start with:           f(x) = d(x) · q(x) + r(x)

    What if d(x) = x - c ? (This is a linear divisor.)    f(x) = (x - c)q(x) + r(x)

    r(x) must be of a degree less than (x - c), so r(x) must be a constant, k

   Let’s see what happens when we plug c in for x to find f(c):

                               f(c) = (c - c)q(c) + k = 0·q(c) + k

    In other words, f(c) is the remainder k.

The Remainder Theorem:

    If polynomial f(x) is divided by x - c, the constant remainder r(x) is f(c).
                              Use it
1.   If f(x) = x3 - 3x2 + x + 5, use the Remainder Theorem to find
     f(-4).

2.   Show that (x - 2) is a factor of f(x) = x3 - 4x2 + 3x + 2.


3.   Find a polynomial f(x) that has zeros at x = 2, -1, and 3.

     (How many could you find? How many that are 3rd degree?
                              Use it
1.   If f(x) = x3 - 3x2 + x + 5, use the Remainder Theorem to find
      f(-4).       Divide f(x) by (x+4).

2.   Show that (x - 2) is a factor of f(x) = x3 - 4x2 + 3x + 2.
               Show that f(2) = 0.

3.   Find a polynomial f(x) that has zeros at x = 2, -1, and 3.
                f(x) = (x-2)(x+1)(x-3)
     (How many could you find? How many that are 3rd degree?
     The answer to both of these is “infinite.” Consider f(x) = (x-
     2)(x+1)(x-3)2 to illustrate the first answer, and f(x) = 3(x-
     2)(x+1)(x-3) to illustrate the second answer.)
                 Part 2
Let’s make it easier.
           Dividing polynomials
Use long division (not the easy part):
                   x  3 2x 4  5x 3  2x  8
           Dividing polynomials
Use long division:
                           2x 3  x 2  3x  11
                     x  3 2x 4  5x 3  2x  8


With a remainder of 25.

What is f(-3)?
            Dividing polynomials
  Now, use synthetic division:
      2x 3  x 2  3x  11 becomes -3 |2 5 0 -2 -8
x  3 2x 4  5x 3  2x  8            |____________

 What is the remainder?

 What is the quotient?
         Use this to find zeros of
               polynomials
What polynomials are represented below?
What is the remainder? The quotient?
What is the full factorization over the set of integers?

What does it all mean?

       -11    | 1 8 -29 44
              |____________
           Use this to find zeros of
                 polynomials
What polynomials are represented below?
What is the remainder? The quotient?
        0               x2 - 3x + 4
What does it all mean?

   For f(x) = x3 + 8x2 - 29x +44

   -11 is a zero/ solution/ root
   (x + 11) is a factor of f(x)

What are the integer factors of f(x)? (x + 11)(x2 - 3x + 4)
 Use this to find other information
        about polynomials
For f(x) = x3 + 8x2 - 29x +44

   use synthetic division to find f(2).
 Use this to find other information
        about polynomials
For f(x) = 2x4 + 5x3 - 3k,

   What is k if f(2) = 39?
  Summary (using different letters)
If f(a) = b, then

         1. b is the remainder of f(x)/(x - a).
         2. (a, b) is on the graph of f(x).
         3. The value of f(x) at a is b.

If b = 0 also, then

         1.   (x - a) is a factor of f(x). Note: linear factor!
         2.   a is a zero of f(x).
         3.   (a, 0) is on the graph of f(x)
         4.   a is an intercept of f(x).
         5.   a is a solution of f(x) = 0.
              Use it one more time
1.   Give a factor of f(x) = x3 - 5x2 - 6x if (6, 0) is on the graph.
     What is another factor?




2.   If (-4, k) is on the graph of x3 - 2x + 2, what is k? How many
     ways could you corroborate your answer?

				
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