Lesson 2 - Mini-Lesson. Section 2.1 Properties of Exponents

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1 Lesson - Mini-Lesson Section.1 Properties of Exponents What is an exponent? An exponent is a number in the superscript location and identifies the number of times the base number is to be multiplied times itself. For example: 3 = In this situation, the exponent 3 is attached to the base. Raising to the 3 rd power indicates that we are to multiply the base times itself a total of 3 times. Here is how we would perform that multiplication: 3 = = 4 = 8 Notes about this property: Zero Exponent Property For any nonzero real number a, a 0 = is undefined a is the base of the exponential expression Problem 1 MEDIA EXAMPLE Zero Exponent Property Simplify the following expressions using the Zero Exponent Property. 5 0 = b) 5 0 = c) 5x 0 = d) 5x 0 = 53

2 Problem WORKED EXAMPLE Zero Exponent Property Simplify the following expressions using the Zero Exponent Property. 0 = 1 b) 0 = ( 1) 0 = ( 1) ( 1) = 1 c) x 0 = ( 1) = Note:x 0 = 1;assumethatx 0 d) ( x) 0 = 1 Note:Thebaseis x;assumethatx 0 Negative Exponent Property For any real number a 0,b 0 : a m = 1 1 a = a m am m a b m = b a m Problem 3 MEDIA EXAMPLE Negative Exponent Property Rewrite each of the following with only positive exponents. Variables represent nonzero quantities. 1 x 3 = b) = x 1 1 c) x = d) 3 = e) 16 = f) 3x 4 = 54

3 Problem 4 WORKED EXAMPLE Negative Exponent Property Rewrite each of the following with only positive exponents. Variables represent nonzero quantities. y 4 = 1 y 4 b) 1 x = x c) 1 1x5 = 5 4x 4 Note:Thecoefficient4doesnotmove d) 4 1 = 1 41 = x5 4 = 1 4 = 1 e) x 5 = x 5 Note:Thecoefficientdoesnotmove. f) = = 3 3 = = 9 3 = 7 Problem 5 You Try Zero Exponent Property and Negative Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 7a 0 = b) 7a 0 = c) 5a 3 = d) 7 a 1 = e) 7a 1 = f) 7 a 1 = 55

4 Multiplication Property: For any real number a, am a n = a m+n Problem 6 MEDIA EXAMPLE Multiplication Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. n 3 n 9 = b) b5 b 4 b = c) 5x 7x 3 = d) 5x + 7x 3 = Problem 7 WORKED EXAMPLE Multiplication Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. q 7 q 5 = q 7+5 = q 1 b) a 4 a 5 b = a 4+( 5 ) b 8 = a 5 b 3 = a b c) = ( 4 7) ( x) ( y 5 y ) 3 = ( 4 7) ( x) ( y ) 5 + ( 3) = ( 8) ( x) ( y ) 4 y 5 7xy 3 d) 4 y5 +7 y 3 = completelysimplified = 8xy 56

5 Raising an Exponent to an Exponent Property: For any real number a, n = a mn am Problem 8 MEDIA EXAMPLE Raising an Exponent to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. x 3 5 = b) x x 3 = Problem 9 WORKED EXAMPLE Raising an Exponent to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. x ( x ) 3 4 = x x 1 = x +1 = x 14 b) 4x x 3 3 = 4x x 3 i 3 = 4x x = 4x +( ) = 4x 0 = 4 1 = 4 57

6 Product and Quotient to an Exponent Property: For any real number a, PRODUCT: n =a n b n n a = an QUOTIENT: ab b b n, b 0 Problem 10 MEDIA EXAMPLE Product to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 5x = b) 5 8ab = c) ( 5n4 3n ) 3 = Problem 11 MEDIA EXAMPLE Quotient to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 5 7 = b) x 5 y 3 4 = Problem 1 WORKED EXAMPLE Product to an Exponent Property and Quotient to an Exponent Property Simplify the following expression. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 1 1a b 4 4 a = 1 = 1 a 1 4 b 8 4 ( b ) 4 = a b 8 58

7 The Division Property: For any nonzero real number a, a m a = n a(m n) Problem 13 MEDIA EXAMPLE The Division Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. x 4 x = 4a 10 b 5 0 b) 6ab = Problem 14 WORKED EXAMPLE The Division Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. y 1 y 6 = y1 6 1 b) y 6 y6 1 = 1 y 1 = y 6 = y 6 1 = 1 y 6 c) 3a 1 b 5 9ab = 1a 1 b 5 = a 1 b 5 6 3ab 6 3a 1 b = a 1 1 b 5 6 = a b 1 = a b = 1 1 3a b Note:Thisproblemshowshorizontalformatwhichisalsoan acceptablewaytowriteyoursolutions. 59

8 Problem 15 You Try Properties of Exponents Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. Show all steps as in the media examples. x 4 = b) x 3 = 3 c) 8g 5g 4 = d) 6 n 3 = e) 3a x 3 y 8 = f) 9xy = 5 60

9 Polynomials and Polynomial Multiplication: Variables raised to exponents and joined by addition form the building blocks, called terms, of algebraic expressions known as polynomials. Some examples are below. Note that the coefficient is the number in front of the variable part of a term. Expression Terms Name Coefficients in Order x 3 1 term:x 3 Monomial 4x 7 terms: 4x, 7 Binomial 4, 7 x + 5 terms: x, 5 Binomial 1, 5 x + 3x 5 3 terms: Trinomial, 3, 5 x, 3x, 5 Polynomial multiplication requires the use of the exponent properties learned previously in this lesson. Some examples are below. Problem 16 Multiply and simplify. MEDIA EXAMPLE Multiplying Polynomials 3x x 4 = b) x x 3 x + 5 = c) 3x x 4 + ( x 4) = Problem 17 Multiply and simplify. x x 4 WORKED EXAMPLE Multiplying Polynomials = x( x) + x( 4) =x 8x b) x( x 4 3x + 6) = ( x) x 4 +( x) ( 3x)+ x = x 5 + ( 3x )+( 6x) = x 5 + 3x 6x ( 6) 61

10 Multiplying Binomials and the FOIL Method: A binomial as a two-term polynomial. When we multiply two binomials, we can use the FOIL method (meaning First, Outside, Inside, Last) to help us keep track of our multiplications. Here is the general form: When we multiply two binomials together, we initially obtain four terms. Usually, two of these will combine resulting in three terms, or, a trinomial. Problem 18 MEDIA EXAMPLE Multiplying Binomials/Higher Order Polynomials Multiply and simplify. x + 3 ( x + 4)= b) d 4 ( 3d + 5)= = d) 3 a c) x x + x 4 = 6

11 Problem 19 WORKED EXAMPLE Multiplying Binomials/Higher Order Polynomials Multiply each set of polynomials below and combine like terms to simplify. ( 3x )( 4x + 3) = ( 3x) ( 4x)+ ( 3x) ( 3)+ ( ) ( 4x)+ = ( 1x )+( 9x)+ ( 8x)+ 6 ( 3) = 1x + 9x 8x 6 = 1x + x 6 b) ( x +1) x + x 5 = ( x) ( x )+( x) ( x)+ ( x) ( 5)+ ( 1) ( x )+( 1) ( x)+ 1 = ( x 3 )+( 1x )+( 5x)+ ( x )+( 1x)+ ( 5) = ( x 3 )+( 1x x )+( 5x +1x)+ ( 5) = ( x 3 )+( x )+( 4x)+ ( 5) = x 3 x 4x 5 ( 5) 63

12 Problem 0 YOU TRY Multiplying Binomials/Higher Order Polynomials Multiply each set of polynomials below and combine like terms to simplify. x 1 ( x + 4) = b) 3x 4 ( 5x + ) = c) x + 5 = d) x 5 = e) x 5 ( x +5) = f) x + ( x ) = g) x 4 ( x + x + 1) = h) x 1 ( x x + 1) = 64

13 Section. Using Properties of Exponents to Evaluate Functions Problem 1 MEDIA EXAMPLE Function Evaluation Given the function, f (x)= x, evaluate each of the following. Show your work. Write final results as ordered pairs. f 5 5 = b) f 6 = c) f 10x + 1 = d) f x 3 = Problem WORKED EXAMPLE Function Evaluation Given f x pairs. = 5 x, evaluate each of the following. Show your work. Write final results as ordered f ( 5) = 5 [Replacexwith 5inf x ( 5) = 5 5 [ ( 5 ) = ( 5) ( 5) = 5] = 1 5 [Reducefraction] 5, 1 5 ] b) = 5 [Replacexwith5xinf x ( 5x) f 5x = 5 5x = 1 5x 1 5x, 5x [( 5x) = ( 5x) ( 5x) = 5x ] [Reducefraction] ] 65

14 Problem 3 You Try Function Evaluation Given f x = 3x, evaluate each of the following. Show your work. Write final results as ordered pairs. f 3 = b) f 5x 3 = c) f x 5 = 66

15 Section.3 Combining Functions Basic Mathematical Operations The basic mathematical operations are: addition, subtraction, multiplication, and division. When working with function notation, these operations will look like this: Addition Subtraction Multiplication Division f ( x ) g( x) f ( x f g( x) g( x) g x f ( x )+ g( x) 0 Problem 4 WORKED EXAMPLE Adding and Subtracting Functions Given f x = x +3x 5 and g x = x +5x +1, determine each of the following. f ( x)+ g( x) = ( x +3x 5)+ ( x +5x +1) = x + 3x 5 x + 5x + 1 = x x + 3x + 5x = x + 8x 4 b) f ( x) g( x) = ( x + 3x 5) ( x + 5x + 1) = x + 3x 5 + x 5x 1 = x + x + 3x 5x 5 1 = 3x x 6 Problem 5 MEDIA EXAMPLE Adding and Subtracting Functions Given f x = 3x + x 1 and g x = x x + 5, determine each of the following. f x + g( x) = b) f x g( x) = Problem 6 YOU TRY Adding and Subtracting Functions Given f x = x + 4 and g x = x 3x + 1, determine each of the following. f x + g( x) = b) f x g( x) = 67

16 Function Multiplication and the Multiplication Property of Exponents When multiplying functions, you will often need to work with exponents. Try to recognize the examples below as being similar to ones completed earlier in the lesson. Problem 7 WORKED EXAMPLE Function Multiplication For each set of functions below, show all work to determine f x g x. Givenf x f ( x) g x = 8x 4 and g( x) = 5x 3, = ( 8x )( 4 5x ) 3 = ( 8 5) ( x 4 x ) 3 = 40x 7 b) Givenf x f ( x) g x = 3x + and g( x) = x 5, = ( 3x +) ( x 5) = ( 3x) ( x)+ ( 3x) ( 5)+ ( ) x = 6x 15x + 4x 10 = 6x 11x 10 + ( 5) Problem 8 MEDIA EXAMPLE Function Multiplication f ( x ) g x Given f x = 3x + and g x = x +3x 1, show all work to determine. Problem 9 YOU TRY Function Multiplication For each of the following, show all work to determine f x g( x). f x = 3x and g( x) = 3x + b) f x = x and g( x) = x 3 4x +5 68

17 Function Division and the Division Property of Exponents When dividing functions, you will also need to work with exponents. Try to recognize the examples below as being similar to ones completed earlier in the lesson. Problem 30 WORKED EXAMPLE Function Division f x For each of the following, determine g x f ( x ) = 15x 15 and g( x) = 3x 9 = 15x f x g x Problem 31 3x 9 15 = 5x 15 9 = 5x 6. Use only positive exponents in your final answer. b) f x MEDIA EXAMPLE Function Division f x For each of the following, determine g x f x = 10x 4 + 3x and g x = 4x 5 and g( x) = x 8 f ( g( x) = 4x 5 x 8 = x 5 8 = x 3 = x 3. Use only positive exponents in your final answer. = x b) f x = 1x 5 + 8x + 5 and g( x) = 4x Problem 3 YOU TRY Function Division f x For each of the following, determine g x f x = 5x 5 4x 7 and g x. Use only positive exponents in your final answer. = 5x 4 b) f x = 0x 6 16x and g( x) = 4x 3 69

18 Working with Functions in Different Forms: Tables and Graphs Functions can be presented in multiple ways including: equations, data sets, graphs and applications. If you understand function notation, then the process for working with functions is the same no matter how the information if presented. Problem 33 MEDIA EXAMPLE Functions in Table Form Functions f (x) and g(x) are defined in the tables below. Find a e below using the tables. f x x g x x f 1 = b) g 9 = c) f 0 + g( 0) = d) g 5 f ( 8) = e) f 0 g( 3) = Problem 34 YOU TRY Functions in Table Form Given the following two tables, complete the third table. Show work in the table cell for each column. The first one is done for you. Show your work the same way as the sample. f x x g x x x f ( 0)+ g( 0) f ( x )+ g( x) = = 10 70

19 Problem 35 YOU TRY Functions in Graph Form Use the graph to determine each of the following. Assume integer answers. g 4 = b) f = c) g 0 = d) If f x = 0, x = e) If g x = 0, x = f) f 1 + g( 1) = g) g 6 f ( 7) = h) f 1 g( ) = i) g 0 f ( 1) = = k) f 1 j) g 6 = g 6 f 1 71

20 Section.4 Applications of Function Operations One of the classic applications of function operations is the forming of the profit function, P x by subtracting the cost function, C x, from the revenue function, R x, as shown below. Profit = Revenue Cost Given functions P x = Profit, R x = Revenue, and C x = Cost: P ( x ) = R( x) C( x), Problem 36 MEDIA EXAMPLE Cost, Revenue, Profit A local courier service estimates its monthly operating costs to be $1500 plus $0.85 per delivery. The revenues are $6 for each delivery. Let x = the number of deliveries in a given month. Write a function, C x, to represent the monthly costs for making x deliveries per month. b) Write a function, R x, to represent the revenue for making x deliveries per month. c) Write a function, P x, that represents the monthly profits for making x deliveries per month. d) Determine algebraically the break-even point for the function P x and how many deliveries you must make each month to begin making money. e) Solve the equation P x = 0 graphically to confirm the break-even point. 7

21 Problem 37 YOU TRY Cost, Revenue, Profit Charlie s Chocolate Shoppe sells their chocolates for $1.80 per piece. The fixed costs to run the Chocolate Shoppe total $450 for the week, and Charlie estimates that each chocolate costs about $0.60 to produce. Charlie estimates that he can produce up to 3,000 chocolates in one week. Write a function, C n, to model Charlie s total weekly costs if he makes n chocolates. b) Write a function, R n, to represent the revenue from the sale of n chocolates. c) Write a function, P n, that represents Charlie s profit from selling n chocolates. d) Interpret the meaning of the statement P 300 = 90. Write your answer as a complete sentence. e) Determine the practical domain and practical range for P n, then use that information to define an appropriate viewing window. Sketch the graph from your calculator. Practical Domain: Practical Range: f) How many chocolates must Charlie sell in order to break even? Show complete work. Write your final answer as a complete sentence. Mark the break-even point on the graph above. 73

Unit 13: Polynomials and Exponents

Unit 13: Polynomials and Exponents Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6: