polynomial division
DESCRIPTION
Polynomial Division. Objective: To divide polynomials by long division 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 ) - PowerPoint PPT PresentationTRANSCRIPT
Polynomial Division
Objective:
To divide polynomials by long division 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
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.