Download 3.3A Polynomial Division

3.3A Dividing Polynomials
Objectives:
A.APR.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A.APR.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form
q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x)
less than the degree of b(x).
For the Board:
You will be able to use long division and synthetic division to divide polynomials.
Bell Work 3.3:
Divide using long division.
1. 161  7
Divide.
6x  15y
3.
3
2. 12.18  2.1
4.
7a 2  ab
a
Anticipatory Set:
Division with polynomials is similar to long division with numbers.
Recall: Long division with numbers.
368  21
17 quotient
divisor 21 368 dividend
21
158
147
11 remainder
Steps:
1.
2.
3.
4.
5.
Little divide.
3/2 = 1
Multiply.
1 x 21 = 21
Subtract.
36 – 21 = 15
Bring down.
158
Repeat until there is nothing to bring down.
11
.
368
It can also be written using the formula
DIVIDEND = QUOTIENT · DIVISOR + REMAINDER
368 = 17 · 21 + 11
This answer is written 17
15/2 = 7
7 x 21 = 147
158 – 147 = 11
done.
Instruction:
Open the book to page 166 and read example 1.
Example: Divide using long division. (-y2 + 2y3 + 25)  (y – 3)
2 y 2  5 y  15
1. Write the dividend in perfect standard form.
y  3 2 y 3  y 2  0 y  25
2. Divide first term of the dividend by the first term of
2y3  6y2
the divisor.
2
3. Multiply this answer by the entire divisor.
5y  0y
4. Subtract
5 y 2  15 y
5. Bring down the next term.
15 y  25
6. Repeat until there is nothing left to bring down.
7. Write any remainder as a fraction over the divisor.
15 y  45
70
70
Solution: 2y 2  5y  15 
y3
This answer can be written using the format P(x) = Q(x) · D(x) + R.
DIVIDEND = QUOTIENT · DIVISOR + REMAINDER
2y3 – y2 + 25 = (2y2 + 5y + 15)(y – 3) + 70
White Board Activity:
Practice: Divide using long division.
a. (15x2 + 8x – 12)  (3x + 1)
5x  1
2
3x  1 15 x  8 x  12
15 x 2  5 x
b. (x2 + 5x – 28)  (x – 3)
x8
2
x  3 x  5 x  28
x 2  3x
3x  12
8 x  28
3x  1
8 y  24
4
 13
5x  1 
13
3x  1
x 8
4
x 3
Write the answers of each of the above division problems using the P(x) = Q(x) · D(x) + R format.
a. (15x2 + 8x – 12) = (5x + 1)(3x + 1) – 13
b. (x2 + 5x – 28) = (x + 8)(x – 3) – 4
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 170 – 171 prob. 2 – 4, 13 – 18, 39 – 48.
For a Grade:
Text: pgs. 170 – 171 prob. 2, 16, 42.