Section 2.3 Polynomial and Synthetic Division

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Section 2.3 Polynomial and Synthetic Division
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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

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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 .
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Long Division.
Check
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Divide.
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Long Division.
Check
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Example

Check
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Division is Multiplication
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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).

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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.

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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
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Long Division.
1
-2
-8
3
3
3
-5
1
1
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The Remainder Theorem

• If a polynomial f(x) is divided by x k, the
  remainder is r f(k).

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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
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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
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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.

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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

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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

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2.3 Homework

• 1-67 odd