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Chapter 8: Exponents and Polynomials

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Chapter 8: Exponents and Polynomials Chapter 8 CHAPTER 8: EXPONENTS AND POLYNOMIALS Chapter Objectives By the end of this chapter, students should be able to: Simplify exponential expressions with positive and/or negative exponents Multiply or divide expressions in scientific notation Evaluate polynomials for specific values Apply arithmetic operations to polynomials Apply special-product formulas to multiply polynomials Divide a polynomial by a monomial or by applying long division CHAPTER 8: EXPONENTS AND POLYNOMIALS ........................................................................................ 211 SECTION 8.1: EXPONENTS RULES AND PROPERTIES ........................................................................... 212 A. PRODUCT RULE OF EXPONENTS .............................................................................................. 212 B. QUOTIENT RULE OF EXPONENTS ............................................................................................. 212 C. POWER RULE OF EXPONENTS .................................................................................................. 213 D. ZERO AS AN EXPONENT............................................................................................................ 214 E. NEGATIVE EXPONENTS ............................................................................................................. 214 F. PROPERTIES OF EXPONENTS .................................................................................................... 215 EXERCISE ........................................................................................................................................... 216 SECTION 8.2 SCIENTIFIC NOTATION ..................................................................................................... 217 A. INTRODUCTION TO SCIENTIFIC NOTATION ............................................................................. 217 B. CONVERT NUMBERS TO SCIENTIFIC NOTATION ..................................................................... 218 C. CONVERT NUMBERS FROM SCIENTIFIC NOTATION TO STANDARD NOTATION .................... 218 D. MULTIPLY AND DIVIDE NUMBERS IN SCIENTIFIC NOTATION ................................................. 219 E. SCIENTIFIC NOTATION APPLICATIONS ..................................................................................... 220 EXERCISE ........................................................................................................................................... 222 SECTION 8.3: POLYNOMIALS ................................................................................................................ 223 A. INTRODUCTION TO POLYNOMIALS ......................................................................................... 223 B. EVALUATING POLYNOMIAL EXPRESSIONS .............................................................................. 225 C. ADD AND SUBTRACT POLYNOMIALS ....................................................................................... 226 D. MULTIPLY POLYNOMIAL EXPRESSIONS ................................................................................... 228 E. SPECIAL PRODUCTS .................................................................................................................. 230 F. POLYNOMIAL DIVISION ............................................................................................................ 231 EXERCISE ........................................................................................................................................... 237 CHAPTER REVIEW ................................................................................................................................. 239 211 Chapter 8 SECTION 8.1: EXPONENTS RULES AND PROPERTIES A. PRODUCT RULE OF EXPONENTS MEDIA LESSON Product rule of exponents (Duration 2:57) View the video lesson, take notes and complete the problems below = ( )( ) = 3 2 5 ∙ Product rule: = + ____________________________! ⋅ Example 1: (2x )(4x )( 3x) Example 2: (5a b )(2a b c ) = ___________________________3 2 = ___________________________3 7 9 2 4 − Warning! The rule can only apply when you have the same base. YOU TRY Simplify: a) 5 5 b) c) (2 )(5 ) 3 10 1 3 2 3 5 2 3 B. QUOTIENT RULE OF EXPONENTS MEDIA LESSON Quotient rule of exponents (Duration 3:12) View the video lesson, take notes and complete the problems below = = 5 ∙ ∙ ∙ ∙ 2 3 Quotient Rule: = ∙ ∙ − _________________________________ Example 1: Example 2: 7 2 7 4 8 5 3 = ___________________________6 = ___________________________ YOU TRY Simplify a) b) c) 13 3 5 2 5 7 5 3 5 3 3 7 2 212 Chapter 8 C. POWER RULE OF EXPONENTS MEDIA LESSON Power rule of exponents (Duration 5:00) View the video lesson, take notes and complete the problems below (ab) =_____________________________ = ________ 3 Power of a product: ( ) = =____________________ =_____________ 3 �� Power of a Quotient: = , if b is not 0. ( ) = _____________________�� = ______ 2 3 Power of a Power: ( ) = ∙ Example 1: (5 ) 4 3 Example 2: 3 2 5 4 � 9 � Warning! It is important to be careful to only use the power of a product rule with multiplication inside parenthesis. This property is not allowed for addition or subtraction, i.e., ( + ) + ( ) YOU TRY ≠ − ≠ − Simplify: c) ( ) a) 3 5 b) 3 7 3 2 4 2 2 � � 2 5 � � d) (4 ) e) f) 2 5 3 3 2 2 4 8 5 � � � 8 � 213 Chapter 8 D. ZERO AS AN EXPONENT MEDIA LESSON Zero as Exponent (Duration 3:51) View the video lesson, take notes and complete the problems below 3=_____________________________________________ 3 Zero Power Rule: = (5 ) (3 )(5 ) Example 1: Example 2: 3 5 0 2 0 0 4 YOU TRY Simplify the expressions completely a) (3x ) b) 2 0 0 6 2 5 3 E. NEGATIVE EXPONENTS MEDIA LESSON Negative Exponents (Duration 4:44) View the video lesson, take notes and complete the problems below = __________________________________________ 3 5 =___________________________________________ = 1 Negative Exponent Rule: = = = − When a and b are not 0. − − � � � � Example 1: Example 2: −5 7 2 −4 −1 −4 5 3 Warning! It is important to note a negative exponent does not imply the expression is negative, only the reciprocal of the base. Hence, negative exponents imply reciprocals. YOU TRY a) b) 3 2 3 −1 5 −1 −4 2 214 Chapter 8 F. PROPERTIES OF EXPONENTS Putting all the rules together, we can simplify more complex expression containing exponents. Here we apply all the rules of exponents to simplify expressions. Exponent Rules Product Quotient Power of Power = = ( ) = + ⋅ − ∙ Power of a Quotient Power of a Product Zero Power ( ) = = = � � Negative Power Reciprocal of Negative Power Negative Power of a Quotient = = = = − − − � � � � MEDIA LESSON Properties of Exponents (Duration 5:00) View the video lesson, take notes and complete the problems below Example 1: (4x y z) (2 ) 5 2 2 4 −2 3 4 Example 2: 2 (3 4 4) −6 −2 �2x y � �x y � −6 4 2 x y YOU TRY Simplify and write your final answers in positive exponents. a) −5 −3 3 −2 b) −2 4 ⋅3 3 −3 �3 � ⋅ −5 3 −4 0 6 2 215 Chapter 8 EXERCISE Simplify. Be sure to follow the simplifying rules and write answers with positive exponents. 1) 4 4 4 2) 4 2 3) 3 4 4 4 2 ∙ ⋅ ⋅ ⋅ 4) 2 4 5) (3 ) 6) (4 ) 4 2 2 3 4 4 2 ⋅ 7) (2 ) 8) (2 ) 9) 5 3 2 2 4 4 4 3 4 10) 11) ( ) 12) 7 2 4 2 3 3 ⋅ 3 3 13) 14) 15) 2 2 3 4 3 3 4 3 3 3 3 16) 17) 3 4 18) ( 2 ) 2 4 2 2 2 4 3 ⋅ ⋅ 4 19) ( 2 ) 20) 2 ( ) 21) 7 5 3 4 2 3 2 4 4 4 2 ⋅ 3 2 3 3 ⋅4 ( ) 22) 23) 24) 3 ( 17 ) 3 4 4 4 3 2 2 2 ⋅2 2 2 4 4 4 3 2 � � � � � � 27) 25) 26) ( ) 5 2 3 2 (2 6 ) 2 2 2 2 ⋅2 2 ⋅2 0 2 4 4 3 2 3 2 2 ⋅ 28) 29) 30) 7 4 ( 2 2) 7 2 4 2 2 ⋅2 2 ⋅� � 2 3 4 4 2 4 ⋅3 2 31) 32) 33) ( 2 2) 7 3 4 2 3 2 2 2 � � �2 � 4 2 2 4 2 2 2 34) 35) 2 (2 ) 36) (3 3 4) 3 −3 2 2 ⋅2 4 −2 3 4 2 3 2 ⋅ −3 3 0 3 ⋅3 37) 38) 39) −1 2 3 4 2 3 −4 2 2 ⋅4 0 4 −1 −4 −4 2 ⋅2 � � 4 ⋅4 40) 4 −2 3 2 −4 2 ⋅�2 � −2 4 216 Chapter 8 SECTION 8.2 SCIENTIFIC NOTATION A. INTRODUCTION TO SCIENTIFIC NOTATION One application of exponent properties is scientific notation. Scientific notation is used to represent really large or really small numbers, like the numbers that are too large or small to display on the calculator. For example, the distance light travels per year in miles is a very large number (5,879,000,000,000) and the mass of a single hydrogen atom in grams is a very small number (0.00000000000000000000000167). Basic operations, such as multiplication and division, with these numbers, would be quite cumbersome. However, the exponent properties allow us for simpler calculations. MEDIA LESSON Introduction of scientific notation (Watch from 0:00 – 9:00) View the video lesson, take notes and complete the problems below 10 =___________ 0 10 =____________ 1 10 =_____________ 2 10 = _____________ 3 10 = _________________________ 100 Avogadro number: 602,200,000,000,000,000,000,000 = ______________________________ MEDIA LESSON Definition of scientific notation (Duration 4:59) View the video lesson, take notes and complete the problems below Standard Form (Standard Notation): _______________________________________________________ Scientific Notation: ____________________________________________________________________ b: _________________________________________ b positive: __________________________________ b negative: _________________________________ Example: Convert to Scientific Notation a) 48,100,000,000 = _________________ b)
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