MTH 209 Week 1

Third

Due for this week…

Homework 1 (on MyMathLab – via the Materials Link) The fifth night after class at 11:59pm.

Read Chapter 6.1-6.4, Do the MyMathLab Self-Check for week 1. Learning team coordination/connections. Complete the Week 1 study plan after submitting

week 1 homework. Participate in the Chat Discussions in the OLS

Slide 2Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Section 5.2

Addition and Subtraction of Polynomials

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Monomials and Polynomials

• Addition of Polynomials

• Subtraction of Polynomials

• Evaluating Polynomial Expressions

Monomials and Polynomials

A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers.Examples of monomials: The degree of monomial is the sum of the exponents of the variables. If the monomial has only one variable, its degree is the exponent of that variable.

The number in a monomial is called the coefficient of the monomial.

3 2 9 88, 7 , , 8 , y x x y xy

Example

Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains and its degree.a. 9y2 + 7y + 4b. 7x4 – 2x3y2 + xy – 4y3 c.Solutiona. The expression is a polynomial with three terms and one variable. The term with the highest degree is 9y2, so the polynomial has degree 2. b. The expression is a polynomial with four terms and two variables. The term with the highest degree is 2x3y2, so the polynomial has degree 5. c. The expression is not a polynomial because it contains division by the polynomial x + 4.

2 38

4x

x

Try Q: 21,23,27 pg 314

Example

State whether each pair of expressions contains like terms or unlike terms. If they are like terms, then add them.a. 9x3, −2x3 b. 5mn2, 8m2nSolutiona. The terms have the same variable raised to the same power, so they are like terms and can be combined.

b. The terms have the same variables, but these variables are not raised to the same power. They are therefore unlike terms and cannot be added.

9x3 + (−2x3) = (9 + (−2))x3 = 7x3

Try Q: 29,31,33 pg 314

Example

Add by combining like terms.

Solution

2 23 4 8 4 5 3x x x x

2 28 3443 5x x x x

2 2 4 8 34 53x x x x

2 23 4 8 4 5 3x x x x

2 4( ) (3 4 )3) (85x x

2 57x x

Try Q: 37,38 pg 314

Example

Simplify.

SolutionWrite the polynomial in a vertical format and then add each column of like terms.

2 2 2 27 3 7 2 2 .x xy y x xy y

2

2

2

2

7 3 7

2 2

yxy

yx y

x

x

2 25 2 5xyx y

Try Q: 41 pg 314

To subtract two polynomials, we add the first polynomial to the opposite of the second polynomial. To find the opposite of a polynomial, we negate each term.

Subtraction of Polynomials

Example

Simplify.

SolutionThe opposite of

3 2 3 25 3 6 5 4 8 .w w w w

3 2 3 25 4 8 is 5 4 8w w w w

3 2 3 25 3 6 5 4 8w w w w

3 2(5 5) (3 4) ( 6 8)w w

3 20 7 2w w

27 2w

Try Q: 57,59,61 pg 314

Example

Simplify.

Solution

2 210 4 5 4 2 1 .x x x x

2

2

10 4 5

4 2 1

x

x

x

x

26 6 6x x

Try Q: 69 pg 315

Example

Write a monomial that represents the total volume of three identical cubes that measure x along each edge. Find the total volume when x = 4 inches.

SolutionThe volume of ONE cube is found by multiplying the length, width and height.

The volume of 3 cubes would be:

3

V x x x

V x

33V x

Example (cont)

Write a monomial that represents the total volume of three identical cubes that measure x along each edge. Find the total volume when x = 4 inches.

SolutionVolume when x = 4 would be:

The volume is 192 square inches.

33V x33(4)

192

V

Try Q: 73 pg 315

Section 5.3

Multiplication of Polynomials

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Multiplying Monomials

• Review of the Distributive Properties

• Multiplying Monomials and Polynomials

• Multiplying Polynomials

Multiplying Monomials

A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. To multiply monomials, we often use the product rule for exponents.

Example

Multiply.a. b.

Solution

a. b.

4 36 3x x 3 4 2(6 )( )xy x y

4 36 3x x

4 3( 6)(3)x

718x

3 4 2(6 )( )xy x y

4 3 26xx y y1 4 3 26x y

5 56x y

Try Q: 9,13 pg 322

Example

Multiply.a. b. c.

Solutiona. b.

c.

3(6 )x 4( 2 )x y (3 5)(7)x

3 36 6( ) 3x x 18 3x

4( ) ( ) ( )( 2 )4 42x y x y

4 8x y

3 5 3( )( ) ( ) ( )757 7x x

21 35x

Try Q: 15,19,21 pg 322

Example

Multiply.a. b.

Solutiona. b.

24 (3 2)xy x y 3 3( )ab a b

24 (3 2)xy x y

23 24 4x yxy xy

212 8xx yy xy

3 3( )ab a b

3 3ab a ab b

4 4a b ab 3 212 8x y xy

Try Q: 23-29 pg 322

Multiplying Polynomials

Monomials, binomials, and trinomials are examples of polynomials.

Example

Multiply.

Solution

( 2)( 4) x x

2 24 4x x xx

2 2( )( ) ( )( )4 )2 ( )4(x xx x x

2 2 4 8x x x

2 6 8x x

Try Q: 39 pg 323

Example

Multiply each binomial.a. b.

Solutiona.

b.

(3 1)( 4)x x 2( 2)(3 1)x x

(3 1)( 4)x x 3 3 4 1 1 4x x x x

23 12 4x x x 23 11 4x x

2( 2)(3 1)x x 2 23 ( 1) 2 3 2 1x x x x 3 23 6 2x x x

Try Q: 51,53,59 pg 323

Example

Multiply.a. b.Solutiona.

b.

24 ( 6 1)x x x 2( 2)( 5 2)x x x

24 4 6 4 1x x x x x

3 24 24 4x x x

24 ( 6 1)x x x

2 5 ( 2) x x x x x

3 2 25 2 2 10 4x x x x x

2( 2)( 5 2)x x x

3 27 8 4x x x

22 2 5 2 2 x x

Try Q: 63,67,69 pg 323

Example

Multiply.

Solution

2 23 ( 3 4 ) ab a ab b

2 233 3 43ab aba ab bab 3 2 2 33 9 12a b a b ab

2 23(3 )4a abab b

Example

Multiply vertically.

Solution

21 (2 3) x x x

22 3

1

x x

x

22 3x x 3 22 3x x x 3 22 4 3x x x

Try Q: 71 pg 323

Section 5.4

Special Products

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Product of a Sum and Difference

• Squaring Binomials

• Cubing Binomials

Example

Multiply.a. (x + 4)(x – 4) b. (3t + 4s)(3t – 4s)Solutiona. We can apply the formula for the product of a sum and difference.

b.

(x + 4)(x – 4)= (x)2 − (4)2

= x2 − 16

(3t + 4s)(3t – 4s) = (3t)2 – (4s)2

= 9t2 – 16s2

Try Q: 7,13,17 pg 329

Example

Use the product of a sum and difference to find 31 ∙ 29.SolutionBecause 31 = 30 + 1 and 29 = 30 – 1, rewrite and evaluate 31 ∙ 29 as follows.

31 ∙ 29 = (30 + 1)(30 – 1)

= 302 – 12

= 900 – 1

= 899

Try Q: 21 pg 329

Example

Multiply.a. (x + 7)2 b. (4 – 3x)2

Solutiona. We can apply the formula for squaring a binomial.

b.

(x + 7)2= (x)2 + 2(x)(7) + (7)2

= x2 + 14x + 49

(4 – 3x)2 = (4)2 − 2(4)(3x) + (3x)2

= 16 − 24x + 9x2

Try Q: 27,29,35,39 pg 330

Example

Multiply (5x – 3)3.Solution

= (5x − 3)(5x − 3)2

= 125x3

(5x – 3)3

= (5x − 3)(25x2 − 30x + 9)

= 125x3 – 225x2 + 135x – 27

– 27 – 150x2+ 45x– 75x2+ 90x

Try Q: 47 pg 330

Example

If a savings account pays x percent annual interest, where x is expressed as a decimal, then after 2 years a sum of money will grow by a factor of (x + 1)2.a. Multiply the expression.b. Evaluate the expression for x = 0.12 (or 12%), and interpret the result. Solutiona. (1 + x)2 = 1 + 2x + x2

b. Let x = 0.12

1 + 2(0.12) + (0.12)2= 1.2544

The sum of money will increase by a factor of 1.2544. For example if $5000 was deposited in the account, the investment would grow to $6272 after 2 years.

Try Q: 75 pg 330

Section 5.6

Dividing Polynomials

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Division by a Monomial

• Division by a Polynomial

Example

Divide.

Solution

5 3

2

6 18

6

x x

x

3

2

56 18

6

x x

x

2 2

5 36 8

6 6

1

x x

x x 3 3x x

Example

Divide.

Solution

25 8 10

5

a a

a

25 8 10

5

a a

a

25 8 10

5 5 5 a a

a a a8 2

5 a

a

Try Q: 17,19,21 pg 348

Example

Divide the expression and then check the result.Solution

Check

5 4 2

3

16 12 8

4

y y y

y

5 4 2

3 3 3

16 12 8

4 4 4

y y y

y y y

2 24 3y y

y

5 4 2

3

16 12 8

4

y y y

y

3 2 24 4 3y y y

y

3 2 3 3 2

4 4 4 3 4y y y y yy

5 4 216 12 8y y y

Try Q: 23 pg 348

Example

Divide and check.Solution

The quotient is 2x + 4 with remainder −4, which alsocan be written as

24 6 8

2 1

x x

x

22 1 4 6 8x x x 2x

4x2 – 2x

8x – 8

8x – 4

−4

+ 4

42 4 .

2 1x

x

Example (cont)

Check: (Divisor )(Quotient) + Remainder = Dividend

(2x – 1)(2x + 4) + (– 4) =2x ∙ 2x + 2x ∙ 4 – 1∙ 2x − 1∙ 4 − 4

= 4x2 + 8x – 2x − 4 − 4

= 4x2 + 6x − 8

It checks.

24 6 8

2 1

x x

x

Try Q: 27 pg 349

Example

Simplify (x3 − 8) ÷ (x − 2).Solution

3 22 0 0 8x x x x x2

x3 – 2x2

2x2 + 0x 2x2 − 4x

4x − 8

+ 2x + 4

04x − 8

The quotient is 2 2 4.x x Try Q: 37 pg 349

Example

Divide 3x4 + 2x3 − 11x2 − 2x + 5 by x2 − 2.Solution

2 4 3 20 2 3 2 11 2 5x x x x x x 3x2

3x4 + 0 – 6x2

2x3 − 5x2 − 2x 2x3 + 0 − 4x

−5x2 + 2x + 5

+ 2x − 5

2x – 5 −5x2 + 0 + 10

The quotient is 22

2 53 2 5 .

2

xx x

x

Try Q: 41 pg 349

Due for this week…

Homework 1 (on MyMathLab – via the Materials Link) The fifth night after class at 11:59pm.

Read Chapter 6.1-6.4 Do the MyMathLab Self-Check for week 1. Learning team planning introductions.

Slide 46Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

End of week 1

You again have the answers to those problems not assigned

Practice is SOOO important in this course. Work as much as you can with MyMathLab, the

materials in the text, and on my Webpage. Do everything you can scrape time up for, first the

hardest topics then the easiest. You are building a skill like typing, skiing, playing a

game, solving puzzles. NEXT TIME: Factoring polynomials, rational

expressions, radical expressions, complex numbers