Division of Polynomials using Long Division method

This lesson is about division of Polynomials

1. REVIEWREVIEW
What is the fiftieth What is the fiftieth
term of the arithmeticterm of the arithmetic
sequence sequence
3, 7, 11, 15…?3, 7, 11, 15…?

2. What type of What type of
sequence is sequence is
described by described by
1, 4, 7, 10, 13, 16…?1, 4, 7, 10, 13, 16…?

3. Which term of the Which term of the
arithmetic sequencearithmetic sequence
4,1,-2,-5,…,is-29 ?4,1,-2,-5,…,is-29 ?

4. DIVISION OF
POLYNOMIALS

BY LONG DIVISION
METHOD

5. (Quick Thinking Only!)
Divide and Write
Example:
19 ÷ 5 = 3 + 4/5
19 = 3(5) + 4
⟷
1. 29 ÷ 5 = _____ ______
⟷
2. 34 ÷ 7 = _____ ______
⟷
3. 145 ÷ 11 = _____ ______
⟷
4. 122 ÷ 7 = ____ _____
⟷
5. 219 ÷15 = ____ _____
⟷

7. 641
23
 xxxx
2
x
1. x
3
DIVIDED x x
2
.
2. Multiply (x-1) by x
2
.
23
xx
2
20x x4
4. Bring down 4x.
5. 2x
2
DIVIDED x 2x
x2
6. Multiply (x-1) by 2x.
xx22
2

x60
8. Bring down -6.
6
9. 6x DIVIDED x
6
66x
0
3. Change sign, Add.
7. Change sign, Add
6
11. Change sign, Add .
10. Multiply (x-1) by 6.
3 2
x x 
2
2 2x x 
6 6x 

8. Long Division.
1583
2
 xxx
x
xx3
2

155x
5
155x
0
)5)(3( xx
Check
1535
2
 xxx
158
2
 xx
2
3x x 
5 15x 

9. Divide.
3
27
3
x
x


3
3 27x x 
3 2
3 0 0 27x x x x   
2
x
3 2
3x x
3 2
3x x 
2
3 0x x
3x
2
3 9x x
2
3 9x x 
9 27x
9
9 27x9 27x 
0

10. Long Division.
824
2
 xxx
x
xx4
2

82x
2
82x
0
)4)(2( xx
Check
824
2
 xxx
82
2
 xx
2
4x x 
2 8x 

12. Example
2026
2
 ppp
p
pp6
2

204p
4
244p
44










6
44
)6()4)(6(
p
ppp
Check
442464
2
 ppp
202
2
 pp
6
202
2


p
pp
6
44


p
2
6p p 
4 24p 
=

13. 202
2
pp
6
202
2


p
pp
6
44
4


p
p
 )6(
6
44
64 

 p
p
pp
4464  pp202
2
pp
)(
)(
)(
)(
)(
xd
xr
xq
xd
xf

)()()()( xrxqxdxf 

14. 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 

15. 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
xr
xq
xd
xf
