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Section 2.3 Polynomial and Synthetic Division

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Section 2.3 Polynomial and Synthetic Division What you should learn How to use long division to divide polynomials by other polynomials How to use synthetic division ... – PowerPoint PPT presentation

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Title: Section 2.3 Polynomial and Synthetic Division


1
Section 2.3 Polynomial and Synthetic Division
2
What you should learn
  • How to use long division to divide polynomials by
    other polynomials
  • How to use synthetic division to divide
    polynomials by binomials of the form
  • (x k)
  • How to use the Remainder Theorem and the Factor
    Theorem

3
x2 times.
1. x goes into x3?
2. Multiply (x-1) by x2.
3. Change sign, Add.
4. Bring down 4x.
5. x goes into 2x2?
2x times.
6. Multiply (x-1) by 2x.
7. Change sign, Add
8. Bring down -6.
9. x goes into 6x?
6 times.
10. Multiply (x-1) by 6.
11. Change sign, Add .
4
Long Division.
Check
5
Divide.
6
Long Division.
Check
7
Example

Check
8
Division is Multiplication
9
The Division Algorithm
  • If f(x) and d(x) are polynomials such that d(x) ?
    0, and the degree of d(x) is less than or equal
    to the degree of f(x), there exists a unique
    polynomials q(x) and r(x) such that
  • Where r(x) 0 or the degree of r(x) is less than
    the degree of d(x).

10
Proper and Improper
  • Since the degree of f(x) is more than or equal to
    d(x), the rational expression f(x)/d(x) is
    improper.
  • Since the degree of r(x) is less than than d(x),
    the rational expression r(x)/d(x) is proper.

11
Synthetic Division
  • Divide x4 10x2 2x 4 by x 3

1
0
-10
-2
4
-3
-3
9
-3
3
-1
1
1
1
-3
12
Long Division.
1
-2
-8
3
3
3
-5
1
1
13
The Remainder Theorem
  • If a polynomial f(x) is divided by x k, the
    remainder is r f(k).

14
The Factor Theorem
  • A polynomial f(x) has a factor (x k) if and
    only if f(k) 0.
  • Show that (x 2) and (x 3) are factors of
  • f(x) 2x4 7x3 4x2 27x 18

2
7
-4
-27
-18
2
4
22
18
36
9
0
2
11
18
15
Example 6 continued
  • Show that (x 2) and (x 3) are factors of
  • f(x) 2x4 7x3 4x2 27x 18

2
7
-4
-27
-18
2
4
22
18
36
9
2
11
18
-3
-6
-15
-9
0
2
5
3
16
Uses of the Remainder in Synthetic Division
  • The remainder r, obtained in synthetic division
    of f(x) by (x k), provides the following
    information.
  • r f(k)
  • If r 0 then (x k) is a factor of f(x).
  • If r 0 then (k, 0) is an x intercept of the
    graph of f.

17
Fun with SYN and the TI-83
  • Use SYN program to calculate f(-3)
  • STAT gt Edit
  • Enter 1, 8, 15 into L1, then 2ndQUIT
  • Run SYN
  • Enter -3

18
Fun with SYN and the TI-83
  • Use SYN program to calculate f(-2/3)
  • STAT gt Edit
  • Enter 15, 10, -6, 0, 14 into L1, then 2ndQUIT
  • Run SYN
  • Enter 2/3

19
2.3 Homework
  • 1-67 odd
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