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Dividing Polynomials
Simple Division -dividing a polynomial by a monomial
2 2 2 26 3 9
3
r s rs r s1.
rs
6r 2s2
3rs
3rs2
3rs
9r 2s
3rs
2rs s 3r
Simplify
3a2b
3ab
6a3b2
3ab
18ab
3ab
a 2a2b 6
2 3 23 6 18
3
a b a b ab2.
ab
Simplify
12x2 y
3x
3x
3x
4xy 1
212 3
3
x y x3.
x
Long Division -divide a polynomial by a polynomial
•Think back to long division from 3rd grade.
•How many times does the divisor go into the dividend? Put that number on top.
•Multiply that number by the divisor and put the result under the dividend.
•Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.
-( )x 3 x 2 5x 24
x
x2 + 3x- 8x
- 8
- 24- 8x - 24
0-( )
x2/x = x
2 5 24
3
x x4.
x
-8x/x = -8
-( )
-( )
-( )h 4 h3 0h2 11h 28
h2
h3 - 4h2
4h2
+ 4h
- 11h4h2 - 16h
5h
h3/h = h2
13 11 28 45. h h h
+ 284h2/h = 4h
+ 5
5h - 2048
48
h 4
5h/h = 5
Synthetic Division -
4 26 : 5 4 6 ( 3)Ex x x x x
To use synthetic division:
•There must be a coefficient for every possible power of the variable.
•The divisor must have a leading coefficient of 1.
divide a polynomial by a polynomial
Step #1: Write the terms of the polynomial so the degrees are in descending order.
5x4 0x3 4x2 x 6
Since the numerator does not contain all the powers of x, you must include a 0 for the x3.
4 2
5 4 6 ( 3)x x x x
Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients.
Since the divisor is x-3, r=3
5x4 0x3 4x2 x 6
5 0 -4 1 63
4 25 4 6 ( 3)x x x x
5
Step #3: Bring down the first coefficient, 5.
3 5 0 - 4 1 6
4 25 4 6 ( 3) x x x x
5
3 5 0 - 4 1 6
Step #4: Multiply the first coefficient by r, so 3 5 15
and place under the second coefficient then add.
15
15
4 2
5 4 6 ( 3)x x x x
5
3 5 0 - 4 1 6
15
15
Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add.
15 3 45
45
41
4 2
5 4 6 ( 3)x x x x
5
3 5 0 - 4 1 6
15
15 45
41
Step #5 cont.: Repeat the same procedure.
123
124
372
378
Where did 123 and 372 come from?
4 2
5 4 6 ( 3)x x x x
Step #6: Write the quotient.The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.
5
3 5 0 - 4 1 6
15
15 45
41
123
124
372
378
4 2
5 4 6 ( 3)x x x x
The quotient is:
5x3 15x2 41x 124 378
x 3
Remember to place the remainder over the divisor.
4 2
5 4 6 ( 3)x x x x
5x 5 21x4 3x3 4x2 2x 2 x 4 Ex 7:
Step#1: Powers are all accounted for and in descending order.
Step#2: Identify r in the divisor.
Since the divisor is x+4, r=-4 .
4 5 21 3 4 2 2
Step#3: Bring down the 1st coefficient. Step#4: Multiply and add.
4 5 21 3 4 2 2
-5
Step#5: Repeat.
20 4 -4 0 8-1 1 0 -2 10
4 3 2 105 2
4x x x
x
5 4 3 25 21 3 4 2 2 4x x x x x x
6x2 2x 4 2x 3 Ex 8:
6x2
2
2x
2
4
2
2x
2
3
2
Notice the leading coefficient of the divisor is 2 not 1.
We must divide everything by 2 to change the coefficient to a 1.
3x2 x 2 x 3
2
3
2 3 1 2
3
9
2
2
2
7
2
21
4
8
4
29
4
26 2 4 2 3x x x
297 43
322
xx
*Remember we cannot have complex fractions - we must simplify.
7 29 13
32 42
xx
7 293
324
2
x
x
7 29
32 4 6
xx
26 2 4 2 3x x x
x3 x 2 2x 7 2x 1 Ex 9:
3 2 2 7 2 1
2 2 2 2 2 2
x x x x
11
2
1
2
7
2 Coefficients
1 1 1 7 1
2 2 2 2
1
2
1
41
4
2
4
1
8
8
8
7
8
7
16
56
16
49
16
3 21 1 7 1
2 2 2 2x x x x
3 2 2 7 2 1x x x x