Dividing Polynomials
Dividing Terms
Example
24 x5
⇒
4 x3
−45 x 7
⇒
9 x3
30 x 6
⇒
12 x 4
Concept
1. Divide or reduce the coefficients.
2. Subtract the exponents (powers) of the same variable.
Examples
42 x8
⇒
−7 x5
35 x 6
⇒
14 x 2
Dividing Polynomials by a Monomial
1. Rewrite using the denominator under each term.
a+b a b
= +
c
c c
2. Divide the terms.
Examples
15 x 2 + 30 x
5x
45 x5 + 12 x 4 − 15 x3
3x 2
18 x 4 − 24 x3
6x2
52 x5 − 30 x3
12 x 2
1
Dividing Polynomials by a Binomial
We will use Long Division to divide.
Quotient
Divisor Dividend
_________
Re mainder
Concept
1. Divide the highest degree term of divisor into highest degree term of dividend.
2. Multiply the Quotient term from step 1 onto the divisor.
3. Subtract the product in step 2 from the dividend.
4. Repeat steps 1, 2, and 3 until degree of remainder is less than degree of divisor.
Example
6x2 + 2x − 5
2x + 4
⇒
2x + 4 6x2 + 2x − 5
Example
6 x3 + 11x 2 − 31x + 16
3x − 2
Example
6 x3 − 44 x + 33
2x + 6
⇒
3 x − 2 6 x3 + 11x 2 − 31x + 16
⇒
2
We can divide by more terms in the divisor.
Example
3 x5 − 4 x 4 − 15 x3 − 8 x + 20
3x 2 + 2 x + 4
Homework: Divide
6 x2 + 8x
1.
2x
2.
14 x 4 − 42 x 2
7 x2
4.
24 x 4 − 54 x3
−18 x 2
7.
75 x3 + 35 x 2 − 25 x
25 x 2
Use Long Division to divide.
5 x 2 + 17 x + 6
10.
x+3
3.
24 x5 + 18 x3
12 x 2
5.
15 x3 + 25 x 2 − 35 x
5x
6.
30 x5 − 18 x3 + 42 x
12 x 2
8.
24 x 4 − 18 x3 − 30 x 2
8x2
9.
85 x 7 − 51x5
17 x3
11.
x2 − 4x + 5
x −5
12.
6 x3 − 4 x 2 + 8 x − 13
2x + 4
13.
6 x 2 + 7 x − 18
3x − 4
14.
6 x 2 − 27
3x − 6
15.
8 x3 + 14 x 2 − 11x + 14
2x + 5
16.
10 x3 + 33 x 2 − 20
5x + 4
17.
8 x3 − 1
2x −1
18.
27 x3 + 8
3x + 2
19.
10x 3 + x 2 − 16
2x − 3
20.
15x 3 + 11x 2 − 2x + 7
5x − 3
21.
16x 4 + 6x − 5
4x − 2
22.
5 x3 + 6 x 2 − 23 x + 12
x2 + 2x − 3
6 x 4 − 8 x3 − x 2 + 20 x − 36
2 x 2 −5
23.
24.
18 x 4 + 15 x3 + 20 x − 32
3x 2 + 4
3