Due for this week. Slide 2. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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1 MTH 09 Week 1

2 Due for this week Homework 1 (on MyMathLab via the Materials Link) The fifth night after class at 11:59pm. Read Chapter , 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 Copyright 009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

3 Section 5. Addition and Subtraction of Polynomials Copyright 013, 009, and 005 Pearson Education, Inc.

4 Objectives Monomials and Polynomials Addition of Polynomials Subtraction of Polynomials Evaluating Polynomial Expressions

5 Monomials and Polynomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. 8, 7 y, x, 8 x y, xy 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.

6 Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains and its degree. a. 9y + 7y + 4 b. 7x 4 x 3 y + xy 4y 3 c. a. The expression is a polynomial with three terms and one variable. The term with the highest degree is 9y, so the polynomial has degree. b. The expression is a polynomial with four terms and two variables. The term with the highest degree is x 3 y, so the polynomial has degree 5. c. The expression is not a polynomial because it contains division by the polynomial x x 3 x 4 Try Q: 1,3,7 pg 314

7 Try Q: 9,31,33 pg 314 State whether each pair of expressions contains like terms or unlike terms. If they are like terms, then add them. a. 9x 3, x 3 b. 5mn, 8m n a. The terms have the same variable raised to the same power, so they are like terms and can be combined. 9x 3 + ( x 3 ) = (9 + ( ))x 3 = 7x 3 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.

8 Add by combining like terms. 3x 4x 8 4x 5x 3 3x 4x 8 4x 5x 3 3x 4x 8 4x 5x 3 3x 4x x 5x ( ) x ( 4 5) x ( ) 7x x 5 Try Q: 37,38 pg 314

9 Simplify. x xy y x xy y Write the polynomial in a vertical format and then add each column of like terms. 7x 3xy 7 x xy y y 5x xy 5y Try Q: 41 pg 314

10 Subtraction of Polynomials 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.

11 Simplify. The opposite of w 3 w w 3 w w 3 4w 8 is 5w 3 4w 8 5w 3 3w 6 5w 3 4w 8 3 (5 5) w (3 4) w ( 6 8) 3 0w 7w 7w Try Q: 57,59,61 pg 314

12 Simplify. x x x x x 4x 5 4x x 1 6x 6x 6 Try Q: 69 pg 315

13 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. The volume of ONE cube is found by multiplying the length, width and height. V x x x V The volume of 3 cubes would be: V 3x x 3 3

14 (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. Volume when x = 4 would be: The volume is 19 square inches. V V 3x 3 3 3(4) 19 Try Q: 73 pg 315

15 Section 5.3 Multiplication of Polynomials Copyright 013, 009, and 005 Pearson Education, Inc.

16 Objectives Multiplying Monomials Review of the Distributive Properties Multiplying Monomials and Polynomials Multiplying Polynomials

17 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.

18 Multiply. 4 3 a. b. 6x 3x 3 4 (6 xy )( x y ) 4 3 a. 6x 3x b. 3 4 (6 xy )( x y ) 4 3 ( 6)(3)x 7 18x 4 3 6xx y y 6x y x y Try Q: 9,13 pg 3

19 Multiply. a. 3(6 x) b. 4( x y) c. a. b. 3( 6 x) x 18 3x (3x 5)(7) 4( x y) 4( x) ( 4)( y) 4x 8y c. ( 3x 5)( 7) 3x( 7) 5( 7) 1x 35 Try Q: 15,19,1 pg 3

20 Multiply. a. 4 xy(3x y ) b. a. 4 xy(3x y ) b. ab a 3 3 ( b ) ab a 3 3 ( b ) 4xy 4 3xy xy 1xx yy 8 xy ab a ab b a b ab x y 8 xy Try Q: 3-9 pg 3

21 Multiplying Polynomials Monomials, binomials, and trinomials are examples of polynomials.

22 Multiply. ( x )( x 4) ( x )( x 4 ) ( x )( x) ( x )() 4 x x x x 4 4 x x x x x Try Q: 39 pg 33

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24 Multiply each binomial. a. (3x 1)( x 4) b. a. ( x )(3x 1) (3x 1)( x 4) 3x x 3x 4 1 x 1 4 b. ( x )(3x 1) 3x 1x x 4 3x 11x 4 x 3 x x ( 1) 3x 1 3 3x x 6x Try Q: 51,53,59 pg 33

25 Multiply. a. 4 x( x 6x 1) b. a. 4 x( x 6x 1) ( x )( x 5x ) 4x x 4x 6x 4x 1 3 4x 4x 4x b. ( x )( x 5x ) x x x x x 5 ( ) x 5x 3 x x x x x x x x Try Q: 63,67,69 pg 33

26 Multiply. 3 ab( a 3ab 4 b ) 3 ab( a 3ab 4b ) 3ab a 3ab 3ab 3ab 4b 3a b 9a b 1ab 3 3

27 Multiply vertically. x 1 (x x 3) x x 3 x 1 x x x x x 3 x x 4x 3 Try Q: 71 pg 33

28 Section 5.4 Special Products Copyright 013, 009, and 005 Pearson Education, Inc.

29 Objectives Product of a Sum and Difference Squaring Binomials Cubing Binomials

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31 Multiply. a. (x + 4)(x 4) b. (3t + 4s)(3t 4s) a. We can apply the formula for the product of a sum and difference. (x + 4)(x 4) = (x) (4) = x 16 b. (3t + 4s)(3t 4s) = (3t) (4s) = 9t 16s Try Q: 7,13,17 pg 39

32 Use the product of a sum and difference to find Because 31 = and 9 = 30 1, rewrite and evaluate 31 9 as follows = (30 + 1)(30 1) = 30 1 = = 899 Try Q: 1 pg 39

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34 Multiply. a. (x + 7) b. (4 3x) a. We can apply the formula for squaring a binomial. (x + 7) = (x) + (x)(7) + (7) = x + 14x + 49 b. (4 3x) = (4) (4)(3x) + (3x) = 16 4x + 9x Try Q: 7,9,35,39 pg 330

35 Multiply (5x 3) 3. (5x 3) 3 = (5x 3)(5x 3) = (5x 3)(5x 30x + 9) = 15x 3 150x + 45x 75x + 90x 7 = 15x 3 5x + 135x 7 Try Q: 47 pg 330

36 Try Q: 75 pg 330 If a savings account pays x percent annual interest, where x is expressed as a decimal, then after years a sum of money will grow by a factor of (x + 1). a. Multiply the expression. b. Evaluate the expression for x = 0.1 (or 1%), and interpret the result. a. (1 + x) = 1 + x 1 + (0.1) x + (0.1) = b. Let x = 0.1 The sum of money will increase by a factor of For example if $5000 was deposited in the account, the investment would grow to $67 after years.

37 Section 5.6 Dividing Polynomials Copyright 013, 009, and 005 Pearson Education, Inc.

38 Division by a Monomial Division by a Polynomial Objectives

39 Divide. 6x 18x 6x x 18 6x x 3 6x 18x 6x 6x 5 3 x 3 3x

40 Divide. 5a 8a 10 5a 5a 8a 10 5a 5a 8a 10 5a 5a 5a 8 a 5 a Try Q: 17,19,1 pg 348

41 Divide the expression check the result. 16y 1y 8y 3 4y y 1y 8y 3 4y 5 4 and then 16y 1y 8y 4y 4y 4y y 3y y Check 4y 4y 3y 3 y 4y 4y 4y 3y 4y y 16y 1y 8y 5 4 Try Q: 3 pg 348

42 Divide and check. 4x 6x 8 x 1 x + 4 x 1 4x 6x 8 4x x 8x 8 8x 4 4 The quotient is x + 4 with remainder 4, which also can be written as 4 x 4. x 1

43 (cont) 4x 6x 8 x 1 Check: (Divisor )(Quotient) + Remainder = Dividend (x 1)(x + 4) + ( 4) = x x + x 4 1 x = 4x + 8x x 4 4 = 4x + 6x 8 It checks. Try Q: 7 pg 349

44 Simplify (x 3 8) (x ). x + x x x 0x 0x 8 x 3 x x + 0x x 4x 4x 8 4x 8 0 The quotient is x x 4. Try Q: 37 pg 349

45 Divide 3x 4 + x 3 11x x + 5 by x. 3x + x x 0x 3x x 11x x 5 3x xx 3 5x x x x 5x + x + 5 5x x 5 x 5 The quotient is 3x x 5. x Try Q: 41 pg 349

46 Due for this week Homework 1 (on MyMathLab via the Materials Link) The fifth night after class at 11:59pm. Read Chapter Do the MyMathLab Self-Check for week 1. Learning team planning introductions. Copyright 009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 46

47 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