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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 to divide polynomials by binomials of the form
(x – k)
• How to use the Remainder Theorem and the Factor Theorem
641 23 xxxx
2x1. x goes into x3? x2 times.
2. Multiply (x-1) by x2.
23 xx 220 x x4
4. Bring down 4x.
5. x goes into 2x2? 2x times.
x2
6. Multiply (x-1) by 2x.
xx 22 2 x60
8. Bring down -6.
69. x goes into 6x?
6
66 x0
3. Change sign, Add.
7. Change sign, Add
6 times.
11. Change sign, Add .
10. Multiply (x-1) by 6.
3 2x x
22 2x x
6 6x
Long Division.
1583 2 xxx
x
xx 32
155 x
5
155 x0
)5)(3( xx
Check
15352 xxx
1582 xx
2 3x x
5 15x
Divide.
3 27
3
x
x
33 27x x
3 23 0 0 27x x x x
2x
3 23x x3 23x x 23 0x x
3x
23 9x x23 9x x 9 27x
9
9 27x 9 27x 0
Long Division.
824 2 xxx
x
xx 42
82 x
2
82 x0
)4)(2( xx
Check
8242 xxx
822 xx
2 4x x
2 8x
Example
2026 2 ppp
p
pp 62
204 p
4
244 p
44
6
44)6()4)(6(
pppp
Check
4424642 ppp
2022 pp
6
2022
p
pp6
44
p
2 6p p
4 24p
=
2022 pp
6
2022
p
pp6
444
pp
)6(6
4464
p
ppp
4464 pp2022 pp
)(
)()(
)(
)(
xd
xrxq
xd
xf
)()()()( xrxqxdxf
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).
)()()()( xrxqxdxf
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.
)(
)()(
)(
)(
xd
xrxq
xd
xf
Synthetic DivisionDivide x4 – 10x2 – 2x + 4 by x + 3
1 0 -10 -2 4-3
1
-3
-3
+9
-1
3
1
-3
1
3
4210 24
x
xxx
3
1
x13 23 xxx
Long Division.
823 2 xxx
x
xx 32
8x
1
3x582)( 2 xxxf
xx 32
3 x
)3(f 8)3(2)3( 2 869
5
1 -2 -83
1
3
1
3
-5
The Remainder Theorem
If a polynomial f(x) is divided by x – k, the remainder is r = f(k).
82)( 2 xxxf
)3(f 8)3(2)3( 2 869
5
823 2 xxx
x
xx 32
8x
1
3x5
xx 32
3 x
The Factor TheoremA 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
2
4
11
22
18
36
9
18
0
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
2
4
11
22
18
36
9
18
-3
2
-6
5
-15
3
-9
0 1827472 234 xxxx)2)(918112( 23 xxxx)3)(2)(352( 2 xxxx)3)(2)(1)(32( xxxx
Uses of the Remainder in Synthetic Division
The remainder r, obtained in synthetic division of f(x) by (x – k), provides the following information.
1. r = f(k)
2. If r = 0 then (x – k) is a factor of f(x).
3. If r = 0 then (k, 0) is an x intercept of the graph of f.
Fun with SYN and the TI-83
• Use SYN program to calculate f(-3)
• [STAT] > Edit
• Enter 1, 8, 15 into L1, then [2nd][QUIT]
• Run SYN
• Enter -3
158)( 2 xxxf )3(f
Fun with SYN and the TI-83
• Use SYN program to calculate f(-2/3)
• [STAT] > Edit
• Enter 15, 10, -6, 0, 14 into L1, then [2nd][QUIT]
• Run SYN
• Enter 2/3
1461015)( 234 xxxxf
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