Warm-Up
Use long division to divide 5 into 3462.
5 34626
30-
46
9
45-
12
2
10-
2
Divisor Dividend
Quotient
Remainder
Warm-Up
Use long division to divide 5 into 3462.
3462 2692
5 5
Dividend
Divisor
Quotient
Remainder
Divisor
Remainders
If you are lucky enough to get a remainder of zero when dividing, then the divisor divides divides evenlyevenly into the dividend
This means that the divisor is a factorfactor of the dividend.
For example, when dividing 3 into 192, the remainder is 0. Therefore, 3 is a factor of 192.
5-3 Dividing Polynomials
Glencoe – Algebra 2Chapter 5: Polynomials
5
Skills: • Divide polynomials using long
division.• Divide polynomials using synthetic
division.
Vocabulary
As a group, define each of these without your book. Give an example of each word and leave a bit of space for additions and revisions.
Quotient Remainder
Dividend Divisor
Divides Evenly Factor
Two Types of Polynomial Division
Polynomial Long Division Synthetic Division
• Always works– Use the normal algorithm
for long division
• Divisor MUST be in the form (x – r)– x cannot be raised to any
power other than one to use synthetic division!
8
Polynomial Long Division
q x
f
x d
x
d x
r x
polynomial
divisor
remainder
divisor
quotient
1. Make sure both the polynomial and divisor are in standard form. (decreasing order of degree)
2. If terms are missing, put them in with 0 coefficients.3. The polynomial goes inside the house and the divisor goes outside. (like
regular long division)4. Focus on the first term and what you’d have to multiply the first term of the
divisor by to get the first term of the polynomial.5. Continue multiplying and subtracting like regular long division until the
remainder is one degree less than the divisor.Glencoe – Algebra 2
Chapter 5: Polynomials
1
497
8 497
9
Example 1Divide 497 by 8.
1 16
1
2
7
48
6
862
8
Glencoe – Algebra 2Chapter 5: Polynomials
Simplify: 3 22 11 10 6 4x x x x
3 22 11 10 6x x x 4x
22x
3 22 8x x
23 10x x
3x
23 12x x 2 6x
2
2 8x
2
2
4x
Ask yourself…x times what gives 2x3?
Answer…x times 2x2 gives 2x3.
Rewrite as follows:
Scrap Paper
22 4x x
3 22 8x x
Change the signs.
Now ask yourself…x times what gives 3x2?Answer…x times 3x gives 3x2.
Scrap Paper
3 4x x
23 12x x
Change the signs.
Now ask yourself…x times what gives -2x?
Answer…x times -2 gives -2x.
Scrap Paper
2 4x
2 8x
Change the signs.
Your exponents must go in descending order.
If you are missing an exponent, put in a zero for that place.
Example: 3 3 21 should be written as: 0 0 1x x x x
Write remainders as fractions.
You must change signs before you add!!!
4 32 3 5 1x x x
210 20 20x x
3 27 14 14x x x
42 x 2 4x 3 4x 4 3 2 2 4 4x x x
4 3 22 3 0 5 1x x x x
12
One last example
4 3
2
Divide 2 3 1 5
by 2 2 .
x x x
x x37x
22x 2 2 2x x 4 32 3 5 1x x x
2 4x 5x
7x
37x 214x 14x210x 9x 1
10
210 x 20x 20
11x 21
11 2
1
x 22 7 10x x 2 2 2x x 2 2 2x x
Glencoe – Algebra 2Chapter 5: Polynomials
4 33 7 11 3x x x x
First, make sure there are no skipped powers. Rewrite with zeros if necessary.
24 33 7 11 30xx x x x
Next, write just the coefficients of the dividend.
3 7 0 1 -11
Then, find out what value makes the divisor equal zero and write that number in the “box”.
-3
Finally, skip a line and draw a line.
Getting the problem set up.
3 7 0 1 -11-3
Getting’ it done.
3
1. Bring down the first number.
2. Multiply the number in the “box” by this number.
3. Place your answer under the next number.
4. Add.
5. Repeat 2-4.
x -9-2
x 66
x -18-17
x
5140
-3
3
-9-2
66
-18-17
5140
3 7 0 1 -11
Now what?!?
Box your last number. This is your remainder.
Your first variable’s exponent will be one less than the dividend’s. The remaining exponents go in descending order.
3 2 403 2 6 17
3x x x
x
3 22 8 5x x x
18
Example 3
3 2Divide 2 8 5
by 3.
x x x
x
3 2 1 8 5265
15
72116
16
22 5 7x x 3x 3x
Glencoe – Algebra 2Chapter 5: Polynomials
3 23 4 28 16x x x
19
Example 4
3 2Factor 3 4 28 16
completely given that 2
is a f actor.
x x x
x
2 3 4 28 16 3610
20
816
0
2x 23 10 8x x
2x 3 2 4x x
Glencoe – Algebra 2Chapter 5: Polynomials
Dividing a Polynomial by a Monomial
Divide each term of the polynomial by the monomial.
Remember to divide coefficients and subtract exponents.
Examples1. Divide 9x2 + 12x – 18 by 3x.
2. (Do in your notes) Divide 32x2 – 16x + 64 by -8x
xx
643
xx
824