Warm-Up

Use long division to divide 5 into 3462.

5 34626

30-

46

9

45-

12

2

10-

2

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

Try these:

2

3

4 3 2 2

3 5 1

1 1

2 4 5 3 2 2 3

x x x

x x

x x x x x x

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

Try these

• Divide using synthetic division

4 2

4 3 2

10 2 4 3

2 7 4 27 18 2

x x x x

x x x x x

Synthetic Division

Class/Homework

• Handout• Oh yes, and one more thing (next slide…)

Dividing a Polynomial by a Monomial

Divide each term of the polynomial by the monomial.

Remember to divide coefficients and subtract exponents.

Divide 12x2 – 20x + 8 by 4x

12x2 – 20x + 8

4x4x 4x

x3 5x

2

Examples1. Divide 9x2 + 12x – 18 by 3x.

2. (Do in your notes) Divide 32x2 – 16x + 64 by -8x

xx

643

xx

824

Practice1. Divide 18x2 + 45x – 36 by

9x

2. Divide 10b3 – 8b2 -5b by – 2b

3. Divide x2 – 8x + 15 by x – 3

4. Divide 5x2 + 3x – 15 by x + 2

Answers:1.

2.

3.

4.

xx

452

2

545 2 bb

5x

2

175

x

x