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Chapter 8

CHAPTER 8: EXPONENTS AND 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 operations to polynomials  Apply special-product formulas to multiply polynomials  Divide a by a or by applying long

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)

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= ( )( ) = 3 2 5 𝑎𝑎 ∙ 𝑎𝑎 𝑎𝑎 𝑎𝑎 𝑎𝑎 𝑎𝑎 𝑎𝑎 𝑎𝑎 Product rule: = 𝒎𝒎 𝒏𝒏 𝒎𝒎+𝒏𝒏 ______!𝒂𝒂 ⋅ 𝒂𝒂 𝒂𝒂

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 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥𝑦𝑦 𝑧𝑧

B. QUOTIENT RULE OF EXPONENTS MEDIA LESSON Quotient rule of exponents (Duration 3:12)

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= = 5 𝑎𝑎 𝑎𝑎 ∙ 𝑎𝑎 ∙ 𝑎𝑎 ∙ 𝑎𝑎 ∙ 𝑎𝑎 2 3 𝑎𝑎 Quotient Rule: = 𝑎𝑎 𝑎𝑎 ∙ 𝑎𝑎 ∙ 𝑎𝑎 𝒎𝒎 𝒂𝒂 𝒎𝒎−𝒏𝒏 𝒏𝒏 ______𝒂𝒂 𝒂𝒂

Example 1: Example 2: 7 2 7 4 8𝑚𝑚 𝑛𝑛 𝑎𝑎 𝑏𝑏 5 3 = ______6𝑚𝑚 𝑛𝑛 = ______𝑎𝑎 𝑏𝑏

YOU TRY

Simplify a) b) c) 13 3 5 2 5 7 5𝑎𝑎 𝑏𝑏 𝑐𝑐 3𝑥𝑥 5 3 3 7 2𝑎𝑎𝑏𝑏 𝑐𝑐 𝑥𝑥 𝑦𝑦 212

Chapter 8 C. POWER RULE OF EXPONENTS MEDIA LESSON Power rule of exponents (Duration 5:00)

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(ab) =______= ______3 Power of a product: ( ) =

𝒎𝒎 𝒎𝒎 𝒎𝒎 𝒂𝒂𝒂𝒂 𝒂𝒂 𝒃𝒃 =______=______𝑎𝑎 3 �𝑏𝑏� Power of a Quotient: = , if b is not 0. 𝒎𝒎 𝒂𝒂 𝒎𝒎 𝒂𝒂 𝒎𝒎 ( ) = ______�𝒃𝒃� 𝒃𝒃= ______2 3 𝑎𝑎 Power of a Power: ( ) = 𝒎𝒎 𝒏𝒏 𝒎𝒎∙𝒏𝒏 𝒂𝒂 𝒂𝒂 Example 1: (5 ) 4 3 Example 2: 3 2 𝑎𝑎 𝑏𝑏 5𝑚𝑚 4 � 9𝑛𝑛 �

 Warning! It is important to be careful to only use the power of a product rule with inside parenthesis. This property is not allowed for or , i.e., ( + ) + ( )𝑚𝑚 𝑚𝑚 𝑚𝑚 YOU TRY 𝑎𝑎 𝑏𝑏 𝑚𝑚 ≠ 𝑎𝑎 𝑚𝑚 𝑏𝑏 𝑚𝑚 𝑎𝑎 − 𝑏𝑏 ≠ 𝑎𝑎 − 𝑏𝑏 Simplify: a) c) ( ) 3 5 b) 𝑥𝑥 3 7 3 2 4 2 2 �𝑦𝑦 � 2 𝑥𝑥 𝑦𝑦𝑧𝑧 �5 �

d) (4 ) e) f) 2 5 3 3 2 2 𝑥𝑥 𝑦𝑦 𝑎𝑎 𝑏𝑏 4𝑥𝑥𝑥𝑥 8 5 �𝑐𝑐 𝑑𝑑 � � 8𝑧𝑧 �

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Chapter 8 D. ZERO AS AN EXPONENT MEDIA LESSON Zero as Exponent (Duration 3:51)

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3=______𝑎𝑎 3 Zero𝑎𝑎 Power Rule: = 𝟎𝟎 (5 ) (3 )(5 ) Example 1: 𝒂𝒂 𝟏𝟏 Example 2: 3 5 0 2 0 0 4 𝑥𝑥 𝑦𝑦𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦

YOU TRY

Simplify the expressions completely a) (3x ) b) 2 0 0 6 2𝑚𝑚 𝑛𝑛 5 3𝑛𝑛

E. NEGATIVE EXPONENTS MEDIA LESSON Negative Exponents (Duration 4:44)

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= ______3 𝑎𝑎 5 𝑎𝑎 =______

1 Negative Exponent Rule: = = = = −𝑚𝑚 𝑚𝑚 𝑚𝑚 When a and b are not 0. −𝒎𝒎 𝟏𝟏 𝑚𝑚 𝒎𝒎 −𝑚𝑚 𝑎𝑎 𝑏𝑏 𝑏𝑏 𝒂𝒂 𝒂𝒂 𝑎𝑎 � � � � 𝑚𝑚 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑎𝑎 Example 1: Example 2: −5 7𝑥𝑥 2 −4 −1 −4 5𝑎𝑎 3 𝑦𝑦𝑧𝑧

 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 𝑎𝑎 𝑏𝑏 𝑐𝑐 −1 5 𝑥𝑥 −1 −4 2𝑑𝑑 𝑒𝑒 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 = = ( ) = 𝒎𝒎 𝒏𝒏 𝒎𝒎+𝒏𝒏 𝒎𝒎 𝒂𝒂 ⋅ 𝒂𝒂 𝒂𝒂 𝒂𝒂 𝒎𝒎−𝒏𝒏 𝒎𝒎 𝒏𝒏 𝒎𝒎∙𝒏𝒏 𝒏𝒏 Power of a𝒂𝒂 Quotient 𝒂𝒂 𝒂𝒂 Power of a Product 𝒂𝒂 Zero Power

( ) = = = 𝒎𝒎 𝒎𝒎 𝒎𝒎 𝒎𝒎 𝒎𝒎 𝒂𝒂 𝒂𝒂 𝟎𝟎 𝒂𝒂𝒂𝒂 𝒂𝒂 𝒃𝒃 � � 𝒎𝒎 𝒂𝒂 𝟏𝟏 Negative Power Reciprocal of𝒃𝒃 Negative𝒃𝒃 Power Negative Power of a Quotient

= = = = −𝒎𝒎 𝒎𝒎 𝒎𝒎 −𝒎𝒎 𝟏𝟏 𝟏𝟏 𝒎𝒎 𝒂𝒂 𝒃𝒃 𝒃𝒃 𝒂𝒂 𝒎𝒎 −𝒎𝒎 � � � � 𝒎𝒎 𝒂𝒂 𝒂𝒂 MEDIA LESSON 𝒂𝒂 𝒃𝒃 𝒂𝒂 𝒂𝒂 Properties of Exponents (Duration 5:00)

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Example 1: (4x y z) (2 ) 5 2 2 4 −2 3 4 Example 2: 2 (3 4 4) −6 −2 𝑥𝑥 𝑦𝑦 𝑧𝑧 �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 4𝑥𝑥 𝑦𝑦 ⋅3𝑥𝑥 𝑦𝑦 3 −3 �3𝑎𝑎𝑏𝑏 � ⋅𝑎𝑎𝑏𝑏 −5 3 −4 0 6𝑥𝑥 𝑦𝑦 2𝑎𝑎 𝑏𝑏

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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 ∙ ⋅ ⋅ 𝑚𝑚 ⋅ 𝑚𝑚𝑚𝑚 4) 2 4 5) (3 ) 6) (4 ) 4 2 2 3 4 4 2 𝑚𝑚 𝑛𝑛 ⋅ 𝑛𝑛𝑚𝑚

7) (2 ) 8) (2 ) 9) 5 3 2 2 4 4 4 𝑢𝑢 𝑣𝑣 𝑎𝑎 3 4 10) 11) ( ) 12) 7 2 4 2 3 3 𝑥𝑥 𝑦𝑦 ⋅ 𝑥𝑥𝑦𝑦 𝑥𝑥𝑥𝑥 3 3 13) 14) 15) 2 2 3 4 3 3𝑛𝑛𝑚𝑚 4𝑥𝑥 𝑦𝑦 3 3 3𝑛𝑛 3𝑥𝑥𝑦𝑦 16) 17) 3 4 18) ( 2 ) 2 4 𝑥𝑥 𝑦𝑦 2 2 2 4 3 𝑥𝑥 ⋅ 𝑥𝑥 𝑢𝑢 𝑣𝑣 ⋅ 𝑢𝑢 4𝑥𝑥𝑥𝑥 19) ( 2 ) 20) 2 ( ) 21) 7 5 3 4 2 3 2 4 4 4 2𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 ⋅ 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥 𝑦𝑦 3 2 3 3𝑥𝑥 𝑦𝑦⋅4𝑥𝑥 𝑦𝑦 ( ) 22) 23) 24) 3 ( 17 ) 3 4 4 4 3 2 2𝑦𝑦 2𝑚𝑚𝑛𝑛 ⋅2𝑚𝑚 𝑛𝑛 2𝑥𝑥 2 4 4 4 3 2𝑥𝑥 𝑦𝑦 � 𝑥𝑥 � � � � 𝑚𝑚𝑛𝑛 � 27) 25) 26) ( ) 5 2 3 2 (2 6 ) 2 2 2𝑦𝑦 2𝑥𝑥𝑦𝑦 ⋅2𝑥𝑥 𝑦𝑦 2𝑥𝑥 𝑦𝑦 𝑧𝑧 ⋅2𝑧𝑧𝑥𝑥 𝑦𝑦 0 2 4 4 3 2 3 2 𝑥𝑥 𝑦𝑦 2𝑥𝑥𝑦𝑦 ⋅𝑦𝑦 𝑥𝑥 𝑧𝑧 28) 29) 30) 7 4 ( 2 2) 7 2 4 2 2𝑏𝑏𝑎𝑎 ⋅2𝑏𝑏 2𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑦𝑦𝑥𝑥 ⋅�𝑦𝑦 � 2 3 4 4 2 4 𝑏𝑏𝑎𝑎 ⋅3𝑎𝑎 𝑏𝑏 𝑏𝑏𝑎𝑎 2𝑦𝑦 31) 32) 33) ( 2 2) 7 3 4 2 3 2 2 2𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑛𝑛 �𝑛𝑛 � �2𝑦𝑦 𝑥𝑥 � 4 2 2 4 2 𝑏𝑏𝑎𝑎 2𝑚𝑚𝑚𝑚 2𝑥𝑥 𝑦𝑦 𝑥𝑥 34) 35) 2 (2 ) 36) (3 3 4) 3 −3 2 2𝑞𝑞 𝑝𝑝 𝑟𝑟 ⋅2𝑝𝑝 4 −2 3 4 2𝑥𝑥 𝑦𝑦 3 2 𝑥𝑥 𝑦𝑦 ⋅ 𝑥𝑥𝑦𝑦 −3 3 0 𝑞𝑞𝑞𝑞𝑝𝑝 3𝑥𝑥 𝑦𝑦 ⋅3𝑥𝑥 37) 38) 39) −1 2 3 4 2 3 −4 𝑢𝑢𝑣𝑣 2𝑎𝑎 𝑏𝑏 2𝑥𝑥𝑦𝑦 ⋅4𝑥𝑥 𝑦𝑦 0 4 −1 −4 −4 2𝑢𝑢 𝑣𝑣 ⋅2𝑢𝑢𝑢𝑢 � 𝑎𝑎 � 4𝑥𝑥 𝑦𝑦 ⋅4𝑥𝑥 40) 4 −2 3 2 −4 2𝑏𝑏 𝑐𝑐 ⋅�2𝑏𝑏 𝑐𝑐 � −2 4 𝑎𝑎 𝑏𝑏 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)

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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)

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Standard Form (Standard Notation): ______

Scientific Notation: ______

b: ______

b positive: ______

b negative: ______

Example: Convert to Scientific Notation

a) 48,100,000,000 = ______b) 0.0000235 = ______

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Chapter 8 Definition Scientific notation is a notation for representing extremely large or small numbers in form of 10 where 1 < a < 10 and b is number of decimal places𝑏𝑏 from the right or left we moved to obtain a. 𝑎𝑎 𝑥𝑥 A few notes regarding scientific notation: • b is the way we convert between scientific and standard notation. • b represents the number of times we multiply by 10. (Recall, multiplying by 10 moves the decimal point of a number one place value.) • We decide which direction to move the decimal (left or right) by remembering that in standard notation, positive exponents are numbers greater than ten and negative exponents are numbers less than one (but larger than zero).

Case 1. If we move the decimal to the left with a number in standard notation, then b will be positive. Case 2. If we move the decimal to the right with a number in standard notation, then b will be negative.

B. CONVERT NUMBERS TO SCIENTIFIC NOTATION MEDIA LESSON Convert standard notation to scientific notation (Duration 1:40)

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Example: Convert to scientific notation

8150000 = 0.00000245 =

YOU TRY

Convert the following number to scientific notation a) 14,200 b) 0.0042

c) How long is a Light-Year? The light-year is a measure of distance, not time. It is the total distance that a beam of light, moving in a straight line, travels in one year is almost 6 trillion (6,000,000,000,000) miles. Express a light year in scientific notation. (Source: NASA Glenn Educational Programs Office https://www.grc.nasa.gov/www/k- 12/aerores.htm)

C. CONVERT NUMBERS FROM SCIENTIFIC NOTATION TO STANDARD NOTATION To convert a number from scientific notation of the form 10 to standard notation, we can follow𝑏𝑏 these rules of thumb. • If b is positive, this means the original 𝑎𝑎number𝑥𝑥 was greater than 10, we move the decimal to the right b times. • If b is negative, this means the original number was less than 1 (but greater than zero), we move the decimal to the left b times.

218

Chapter 8 MEDIA LESSON Convert scientific notation to standard notation (Duration 2:22)

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Example: Rewrite in standard notation (decimal notation)

a) 7.85 × 10 b) 1.6 × 10 6 −4

YOU TRY Covert the following scientific notation to standard notation

a) 3.21 × 10 b) 7.4 × 10 5 −3

D. MULTIPLY AND DIVIDE NUMBERS IN SCIENTIFIC NOTATION Converting numbers between standard notation and scientific notation is important in understanding scientific notation and its purpose. Next, we multiply and divide numbers in scientific notation using the exponent properties. If the immediate result is not written in scientific notation, we will complete an additional step in writing the answer in scientific notation.

Steps for multiplying and dividing numbers in scientific notation

Step 1. Rewrite the factors as multiplying or dividing a-values and then multiplying or dividing 10b values.

Step 2. Multiply or divide the a values and apply the product or quotient rule of exponents to add or subtract the exponents, b, on the base 10s, respectively.

Step 3. Be sure the result is in scientific notation. If not, then rewrite in scientific notation.

MEDIA LESSON Multiply and divide scientific notation (Duration 2:47)

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• Multiply/ Divide the ______• Use ______on the 10s

Example:

a) (3.4 × 10 )(2 7 × 10 ) . × b) 5 −2 . × 4 ⋅ 5 32 10 −3 1 9 10

MEDIA LESSON Multiply scientific notations with simplifying final answer step (Duration 3:47)

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Example: a) (1.2 × 10 )(5.3 × 10 ) b) (9 × 10 )(7 × 10 ) 4 3 1 9 219

Chapter 8 MEDIA LESSON Divide scientific notations with simplifying final answer step (Duration 3:44)

View the video lesson, take notes and complete the problems below × . × a) b) × 12 . × 7 7 10 2 4 10 7 2 2 10 4 8 10

YOU TRY Multiply or divide

a) (2.1 10 )(3.7 10 ) . b) −7 5 . 4 𝑥𝑥 𝑥𝑥 4 96 𝑥𝑥 10 −3 3 1 𝑥𝑥 10

c) (4.7 10 )(6.1 10 ) d) (2 × 10 )(8.8 × 10 ) −3 9 6 5 𝑥𝑥 𝑥𝑥

. × . e) f) × 5 . −3 8 4 10 2 014 𝑥𝑥 10 2 −7 7 10 3 8 𝑥𝑥 10

E. SCIENTIFIC NOTATION APPLICATIONS MEDIA LESSON Scientific notation application example 1 (Duration 2:36)

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Example 1: There were approximately 50,000 finishers of the 2015 New York City Marathon. Each finisher ran a distance of 26.1 miles. If you add together the total number miles ran by all the runners, how many times around the earth would the marathon runners have ran? Assume the circumference of the earth to be approximately 2.5 x 104 miles.

Total distance = ______

______

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Chapter 8 MEDIA LESSON Scientific notation application example 2 (Duration 3:24)

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Example 2: If a computer can conduct 400 trillion operations per second, how long would it take the computer to perform 500 million operations? 400 trillion = ______500 million = ______Number of Operations: ______Rate of Operations: ______YOU TRY

a) It takes approximately 3.7 x 104 hours for the light on Proxima Centauri, the next closet star to our sun, to reach us from there. The speed of light is 6.71 x 108 miles per hours. What is the distance from there to earth? Given distance = rate x time. Express your answer in scientific notation

By ESO/Pale Red Dot - http://www.eso.org/public/images/ann16002a/, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=46463949

a) If the North Pole and the South Pole ice were to melt, the north polar ice would make essentially no contribution since it is float ice. However, the south polar ice would make a considerable contribution since it overlays the Antarctic land mass and is not float ice. If Antarctic ice melted, it would become approximately 1.5 x 109 gallons of water. If it takes roughly, 6 x 106 gallons of water to fill 1 foot of the earth, estimate how many feet the earth’s oceans would rise? Express your answer in the standard form. (Source: NASA Glenn Educational Programs Office https://www.grc.nasa.gov/www/k-12/aerores.htm)

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Chapter 8

EXERCISE Write each number in scientific notation 1) 885 2) 0.081 3) 0.000039 4) 0.000744 5) 1.09 6) 15,000 Write each number in standard notation.

7) 8.7 × 10 8) 9 × 10 9) 2 × 10 5 −4 0 10) 2.56 × 10 11) 5 × 10 12) 6 × 10 2 4 −5

Simplify. Write each answer in scientific notation.

13) (7 × 10 )(2 × 10 ) 14) (5.26 × 10 )(3.16 × 10 ) 15) (2.6 × 10 )(6 × 10 ) 1 3 5 2 −2 −2

16) (3.6 × 10 )(6.1 × 10 ) 17) (6.66 × 10 )(4.23 × 10 ) 18) (3.15 × 10 )(8.8 × 10 ) 0 −3 −4 1 3 −5

. × . × . × 19) 20) 21) . × 6 × 6 . × −6 4 81 10 5 33 10 4 08 10 2 3 −4 9 62× 10 .2 ×10 5. 1× 10 22) 23) 24) × 4 × −3 . × −6 9 10 3 22 10 1 3 10 −2 −6 0 3. 10× 7× 10 6. 5× 10 25) 26) 27) . × 3 . × 6 × 5 5 8 10 5 10 8 4 10 −3 2 −2 Scientific5 8 Notation10 Applications 2 5 10 7 10 (Source: NASA Glenn Educational Programs Office https://www.grc.nasa.gov/www/k-12/aerores.htm)

28) The mass of the sun is 1.98 x 1,033 grams. If a single proton has a mass of 1.6 x 10-24 grams, how many protons are in the sun?

29) Pluto is located at a distance of 5.9 x 1014 centimeters from Earth. At the speed of light (2.99 x 1010 cm/sec), approximately how many hours does it take a light signal (or radio message) to travel to Pluto and return? Write your answer standard form.

30) The planet Osiris was discovered by astronomers in 1999 and is at a distance of 150 light-years (1 light-year = 9.2 x 1012 kilometers). a) How many kilometers is Osiris from earth? Express your answer in scientific notation. b) If an interstellar probe were sent to investigate this world up close, traveling at a maximum speed of 700 km/sec or 7 x 102 km/sec, how many seconds would it take to reach Osiris? c) There is about 3.15 x 106 seconds in a year. How many years would it take to reach Osiris?

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Chapter 8

SECTION 8.3: POLYNOMIALS A. INTRODUCTION TO POLYNOMIALS MEDIA LESSON Algebraic Expression Vocabulary (Duration 5:52)

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Definitions Terms: Parts of an algebraic expression separated by addition or subtraction (+ or −) symbols. Constant Term: A number with no variable factors. A term whose value never changes. Factors: Numbers or variable that are multiplied together Coefficient: The number that multiplies the variable.

Example 1: Consider the algebraic expression 4 + 3 22 + 17 5 4 2 a. List the terms: ______𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 b. Identify the constant term. ______

Example 2: Complete the table below 1 2 4 2 5 𝑟𝑟 List of Factors − 𝑚𝑚 −𝑥𝑥 𝑏𝑏ℎ

Identify the Coefficient

Example 3: Consider the algebraic expression 5 8 + 7 a. How many terms are there? ______4 3 2 𝑦𝑦 𝑦𝑦 − 𝑦𝑦 𝑦𝑦 − 4 − b. Identify the constant term. ______c. What is the coefficient of the first term? ______d. What is the coefficient of the second term ______e. What is the coefficient of the third term? ______f. List the factors of the fourth term. ______

YOU TRY

Example 3: Consider the algebraic expression 3 + 4 2 + 8 5 4 a. How many terms are there? ______𝑥𝑥 𝑥𝑥 − 𝑥𝑥 b. Identify the constant term. ______c. What is the coefficient of the first term? ______d. What is the coefficient of the second term ______e. What is the coefficient of the third term? ______f. List the factors of the third term. ______

223

Chapter 8 MEDIA LESSON Introduction to polynomials (Duration 7:12)

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Definitions

Polynomial: An algebraic expression composed of the sum of terms containing a single variable raised to a non-negative exponent.

Monomial: A polynomial consisting of one term, example: ______Binomial: A polynomial consisting of two terms, example: ______Trinomial: A polynomial consisting of three terms, example: ______Leading Term: The term that contains the highest power of the variable in a polynomial, example: ______Leading Coefficient: The coefficient of the leading term, example: ______Constant Term: A number with no variable factors. A term whose value never changes. Example: ______Degree: The highest exponent in a polynomial , example: ______

Example 1: Complete the table below

Name Leading Polynomial Constant Term Degree Coefficient 24 + + 5 6 2 2 +𝑎𝑎 𝑎𝑎 2 8 3 2 𝑚𝑚 5 𝑚𝑚+ − 𝑚𝑚7 − 2 3 𝑥𝑥 2 𝑥𝑥+ 4−

− 4𝑥𝑥 3 𝑥𝑥 YOU TRY Complete the table below

Name Leading Polynomial Constant Term Degree Coefficient 2 + 8 2 𝑛𝑛 −7 𝑛𝑛 2 6 𝑦𝑦 7

𝑥𝑥 −

224

Chapter 8 MEDIA LESSON Introduction to polynomials 2 (Duration 2:58)

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Given: 9 + 7 5 4 3 2 1st term:𝑦𝑦 ______𝑦𝑦 − − 𝑦𝑦 Degree:______Coefficient:______2nd term: ______Degree:______Coefficient:______3rd term: ______Degree:______Coefficient:______4th term: ______Degree:______Coefficient:______

Leading coefficient: ______Degree of leading term: ______Degree of polynomial: ______Write the polynomial in descending order: ______(Or write the polynomial in the standard form)

Standard form of a polynomial The standard form of a polynomial is where the polynomial is written with descending exponents. For example: Rewrite the polynomial in standard form and identify the coefficients, variable terms, and degree of the polynomial 12 + + 2

The standard form of the above2 polynomial3 is 12 + 2. − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 3 2 The coefficients are 1; −12; −1, and 2; the variable terms are , 12𝑥𝑥 −, 𝑥𝑥. The− 𝑥𝑥 degree of the polynomial is 3 because that is the highest degree of all terms. 3 2 𝑥𝑥 − 𝑥𝑥 −𝑥𝑥 YOU TRY

Write the following polynomials in the descending order or in standard form: a) 3 9 + 2 + 7 3 + b) 5 5 + 3 4 2 3 6 2 4 2 4 3 7 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑚𝑚 − 𝑚𝑚 − 𝑚𝑚 − 𝑚𝑚 B. EVALUATING POLYNOMIAL EXPRESSIONS MEDIA LESSON Evaluating algebraic expressions (Duration 7:48)

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To evaluate an algebraic or variable expression, ______the value of the variables into the expression. Then evaluate using the order of operations.

Example 1: If we are given 5 12 and = 17 Example 2: Let = 3, = 7, = 2 we can evaluate. Evaluate 3 + 7 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑦𝑦 𝑧𝑧 − 5 12 𝑥𝑥 − 𝑦𝑦

= 5 ( ___ ) – 12 𝑥𝑥 − = ______Evaluate 2 + 5 2 3 𝑥𝑥 𝑦𝑦 − 𝑧𝑧 225

Chapter 8 Example 4: Let = 3, = 5. Evaluate 4 3 Example 3: Let = 3 . Evaluate 8 + 2 2 9 𝑥𝑥 𝑦𝑦 − 𝑥𝑥 − 𝑦𝑦 − 𝑦𝑦 𝑦𝑦

= 2 Example 6: Let = 2, = 3. Evaluate Example 5: Let . 2 2 Evaluate 3 + 2 + 9 𝑥𝑥 𝑦𝑦 − 𝑥𝑥 𝑦𝑦 − 2 3 2 2 𝑥𝑥 −2𝑦𝑦 𝑥𝑥 − 𝑥𝑥 𝑥𝑥

YOU TRY

a) Evaluate 2 4 + 6 when = 4 b) Evaluate + 2 + 6 when = 3 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − −𝑥𝑥 𝑥𝑥 𝑥𝑥

C. ADD AND SUBTRACT POLYNOMIALS

Combining like terms review MEDIA LESSON Combine like terms 1 (Duration 4:36)

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Definition Like terms: Two or more terms are like terms if they have the same variable or variables with the same exponents.

Which of these terms are like terms? 2 , 2 , 2 , 7 , 49, 0 , 3 3 2 2 Like terms: ______− 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥 𝑦𝑦

Like terms: ______

To combine like terms, we ______. The variable factors ______.

Example: Simplify each polynomials, if possible.

a) 4 7 b) 2 + 4 + 2 9 5 + 2 3 3 2 2 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 𝑦𝑦 − 𝑦𝑦 − 𝑦𝑦 − 𝑦𝑦

226

Chapter 8 MEDIA LESSON Combine like terms 2 (Duration 2:15)

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Combine like terms

a) + 3 + 4 b) 7 4 + 2 + 9 2 2 2 𝑥𝑥 𝑦𝑦 𝑥𝑥𝑦𝑦 𝑥𝑥 𝑦𝑦 − 𝑚𝑚 − 𝑚𝑚

YOU TRY Combine like terms

a) 5 + 2 5 3 + 1 b) 3 2 + 6 + 3 5 3 2 2 2 2 2 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥𝑦𝑦 − 𝑥𝑥 𝑦𝑦 − 𝑥𝑥𝑦𝑦 −

c) 3 + 9 5 3 + 5 d) 3 3 + 5 + 7 10 2 2 2 2 2 2 2 2 2 𝑥𝑥 𝑦𝑦𝑦𝑦 𝑥𝑥 − 𝑥𝑥𝑦𝑦 𝑧𝑧 − 𝑦𝑦 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑦𝑦 − 𝑎𝑎𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦 Add and subtract polynomials MEDIA LESSON Add and subtract polynomials (Duration 3:53)

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To add polynomials: ______

To subtract polynomials: ______

a) (5 7 + ) + (2 + 5 14) b) (3 4 + 7) (8 + 9 2) 2 2 3 3 𝑥𝑥 − 𝑥𝑥 𝑞𝑞 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 −

MEDIA LESSON Add and subtract polynomials (Duration 5:04)

View the video lesson, take notes and complete the problems below c) (2 6 12 4) (11 + 8 + 2 + 6) 5 3 2 5 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − − 𝑥𝑥 𝑥𝑥 𝑥𝑥

d) ( 9 6 11 + 2) ( 9 8 + 4 + 2 ) 3 2 4 3 2 − 𝑦𝑦 − 𝑦𝑦 − 𝑥𝑥 − − 𝑦𝑦 − 𝑦𝑦 𝑥𝑥 𝑥𝑥

227

Chapter 8 YOU TRY

Perform the operation below. a) (4 2 + 8) + (3 9 11) b) (5 2 + 7) (3 + 6 4) 3 3 2 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 −

c) (2 4 + 3) + (5 6 + 1) ( 9 + 8) 2 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥

D. MULTIPLY POLYNOMIAL EXPRESSIONS 1. Distributive property review MEDIA LESSON Distribute property review (Duration 6:08)

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Distributive Property ( + ) = +

= 2 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 = 3 𝑎𝑎 = 4 Example:𝑏𝑏 Use the distributive property to expand each of the following expressions a) 𝑐𝑐5(2 + 4) b) 3( 2 + 7) 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥

c) (5 8) 4 d) − 𝑥𝑥 − 2 𝑥𝑥 1 5 �4 − 3�

YOU TRY

Use the distributive property to expand each of the following expressions. a) 4( 5 + 9 3) b) 7( 2 + 2) 2 2 − 𝑥𝑥 𝑥𝑥 − − − 𝑚𝑚 𝑚𝑚 −

2. Multiply a polynomial by a monomial MEDIA LESSON Multiply a polynomial by a monomial (Duration 2:46)

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To multiply a monomial by a polynomial: ______

Example 1: 5 (6 2 + 5) Example 2: 3 (6 + 2 7) 2 2 4 3 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 228

Chapter 8 YOU TRY Multiply

a) 4 (5 2 + 5) b) 2 (3 4 ) 3 2 3 2 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑏𝑏 − 𝑎𝑎

3. Multiplying with binomials MEDIA LESSON Multiply binomials (Duration 4:27)

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To multiply a by a binomial: ______

______

This process is often called ______, which stands for ______

Example:

a) (4 2)(5 + 1) b) (3 7)(2 8)

𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 −

YOU TRY Multiply

a) (3 + 5)( + 13) b) (4 + 7 )(3 2 )

𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑥𝑥 − 𝑦𝑦

4. Multiply with trinomials MEDIA LESSON Multiply with trinomials (Duration 5:00)

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Multiplying trinomials is just like ______, we just have to ______.

Example: a) (2 4)(3 5 + 1) 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥

b) (2 6 + 1)(4 2 6) 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 229

Chapter 8 YOU TRY

Multiply a) (2 5)(4 7 + 3) 2 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥

b) (5 + 10)(3 10 6) 2 2 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 −

E. SPECIAL PRODUCTS There are a few shortcuts that we can take when multiplying polynomials. If we can recognize when to use them, we should so that we can obtain the results even quicker. In future chapters, we will need to be efficient in these techniques since multiplying polynomials will only be one of the steps in the problem. These two formulas are important to commit to memory. The more familiar we are with them, the next two chapters will be so much easier.

1. Difference of two squares MEDIA LESSON Difference of two squares (Duration 2:33)

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Sum and difference

( + )( ) = ______

𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 = ______Sum and difference shortcut: ( + )( ) = ______

𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 Example: a) ( + 5)( 5) b) (6 2)(6 + 2)

𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥

YOU TRY

Simplify: a) (3 + 7)(3 7) b) (8 )(8 + ) 2 2 𝑥𝑥 𝑥𝑥 − − 𝑥𝑥 𝑥𝑥

230

Chapter 8 2. Perfect square trinomials Another shortcut used to multiply binomials is called perfect square trinomials. These are easy to recognize because this product is the square of a binomial. Let’s take a look at an example.

MEDIA LESSON Perfect Square (Duration 3:40)

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Perfect square

( + ) = ______2 𝑎𝑎 𝑏𝑏 Perfect square shortcut: ( + ) = ______2 𝑎𝑎 𝑏𝑏 Example: a) ( 4) b) (2 + 7) 2 2 𝑥𝑥 − 𝑥𝑥

YOU TRY

Simplify: a) ( 5) b) (2 + 9) 2 2 𝑥𝑥 − 𝑥𝑥 c) (3 7 ) d) (6 2 ) 2 2 𝑥𝑥 − 𝑦𝑦 − 𝑚𝑚

F. POLYNOMIAL DIVISION Dividing polynomials is a process very similar to of whole numbers. Before we look at long division with polynomials, we will first master dividing a polynomial by a monomial.

1. Polynomial division with

MEDIA LESSON Dividing polynomials by monomials - Separated fractions method (Duration 8:14)

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We divide a polynomial by a monomial by rewriting the expression as separated fractions rather than one fraction. We use the fact: = + Example: 𝑎𝑎+𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑐𝑐 𝑐𝑐 a) b) 8 −6w 3𝑥𝑥−6 3 30ω 2

231

Chapter 8

c) d) 3 2 2 6𝑥𝑥 +2𝑥𝑥 −4 20𝑎𝑎 +35𝑎𝑎−4 2 4𝑥𝑥 −5𝑎𝑎

YOU TRY

Simplify

a) b) 5 4 3 2 3 2 9𝑥𝑥 +6𝑥𝑥 −18𝑥𝑥 −24𝑥𝑥 8𝑥𝑥 +4𝑥𝑥 −2𝑥𝑥+6 2 2 3𝑥𝑥 4𝑥𝑥

MEDIA LESSON Long division review (Duration 3:55)

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Long division review

5 2632 Long division steps: 1. ______

2. ______

3. ______

4. ______5. ______This method may seem elementary, but it isn’t the arithmetic we want to review, it is the method. We use the same method as we did in arithmetic, but now with polynomials. MEDIA LESSON Dividing polynomials by monomials – Long division method (Duration 5:00)

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Example:

a) 5 3 5𝑥𝑥 +18𝑥𝑥−9𝑥𝑥 2 3𝑥𝑥

232

Chapter 8

b) 6 5 4 15𝑎𝑎 −25𝑎𝑎 +5𝑎𝑎 4 5𝑎𝑎

YOU TRY

Divide using the long division method

a) 6 4 3 8𝑥𝑥 + 20𝑥𝑥 + 4𝑥𝑥 3 4𝑥𝑥

b) 4 3 2 𝑛𝑛 − 𝑛𝑛 + 𝑛𝑛 𝑛𝑛

c) 4 3 2 12𝑥𝑥 − 24𝑥𝑥 + 3𝑥𝑥 6𝑥𝑥

233

Chapter 8 2. Polynomial division with polynomials MEDIA LESSON Divide a polynomial by a polynomial (Duration 5:00)

View the video lesson, take notes and complete the problems below Polynomial division with polynomials

On division step, only focus on the ______

Example 1: Divide 3 2 𝑥𝑥 −2𝑥𝑥 −15𝑥𝑥+30 𝑥𝑥+4

Example 2: Divide 3 4𝑥𝑥 −6𝑥𝑥+12+8 2𝑥𝑥+1

YOU TRY

a) 2 = 𝑥𝑥 +8𝑥𝑥+12 𝑥𝑥+1

b) = 3 2 3𝑥𝑥 −5𝑥𝑥 −32𝑥𝑥+7 𝑥𝑥−4

234

Chapter 8

c) 3 2 = 6𝑥𝑥 −8𝑥𝑥 +10𝑥𝑥+103 2𝑥𝑥+4

MEDIA LESSON Divide a polynomial by a polynomial - rewriting the remainder as an expression (Duration 5:10)

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Example: Divide 3 2 𝑥𝑥 +8𝑥𝑥 −17𝑥𝑥−15 𝑥𝑥+3

YOU TRY Divide the polynomials and write the remainder as an expression

a) 2 = 𝑥𝑥 −5𝑥𝑥+7 𝑥𝑥−2

b) 3 2 = 𝑥𝑥 −4𝑥𝑥 −6𝑥𝑥+4 𝑥𝑥−1

235

Chapter 8 3. Polynomial division with missing terms Sometimes when dividing with polynomials, there may be a missing term in the dividend. We do not ignore the term, we just write in 0 as the coefficient.

MEDIA LESSON Polynomial division with missing terms (Duration 5:00)

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Divide polynomials – Missing terms The exponents must ______. If one is missing, we will add ______.

Example 1: 3 3𝑥𝑥 −50𝑥𝑥+4 𝑥𝑥−4

Example 2: 3 2 2𝑥𝑥 +4𝑥𝑥 +9 𝑥𝑥+3

YOU TRY

a) = 3

2𝑥𝑥 −4𝑥𝑥+42 𝑥𝑥+3

b) 3 2 = 3𝑥𝑥 −3𝑥𝑥 +4 𝑥𝑥−3

236

Chapter 8 EXERCISE Evaluate the expression for the given value. Show your work.

1. + 6 21 when = 4 2. 3 11 when = 6 3 2 2 3. −𝑎𝑎 −7 𝑎𝑎 + 15𝑎𝑎 − 20 when𝑎𝑎 =−2 4. 𝑛𝑛 − 9𝑛𝑛 −+ 23 21𝑛𝑛 when− = 5 3 2 3 2 5. 𝑛𝑛 5− 𝑛𝑛 11 𝑛𝑛 9− 5𝑛𝑛 when = 2 6. 𝑛𝑛 − 5𝑛𝑛 +𝑛𝑛13− when =𝑛𝑛 1 4 3 2 4 3 7. − +𝑛𝑛 9− + 23𝑛𝑛 when− 𝑛𝑛 −=𝑛𝑛 −3 𝑛𝑛 8. 𝑥𝑥 −+𝑥𝑥 − 𝑥𝑥 + 11 when 𝑥𝑥 = 6 2 3 2 9. 𝑥𝑥 𝑥𝑥6 + 24𝑥𝑥 when− = 1 10. −𝑥𝑥+ 𝑥𝑥 +−2 𝑥𝑥 + 13 + 5 𝑥𝑥 when = 3 4 3 2 4 3 2 −𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 Simplify. Write the answer in standard form. Show your work.

11. (5 5 ) (8 8 ) 12. (3 ) (2 7 ) 4 4 2 3 3 2 13. (8𝑝𝑝 −+ 𝑝𝑝) −(3 𝑝𝑝 −4 𝑝𝑝) 14. (1𝑛𝑛+ 5− 𝑛𝑛) −(1 𝑛𝑛8− ) 𝑛𝑛 4 4 3 3 15. (5𝑛𝑛 +𝑛𝑛6 −) + 𝑛𝑛(8− 3𝑛𝑛 5 ) 16. (3 + 𝑝𝑝) +−(7 +−2 𝑝𝑝+ ) 4 3 3 4 4 4 17. (8𝑛𝑛 + 1)𝑛𝑛 (5 − 6𝑛𝑛 +− 2)𝑛𝑛 18. (2 +𝑏𝑏2 ) (3 𝑏𝑏 6𝑏𝑏 + 3) 3 4 3 4 2 19. (4𝑥𝑥 3 −2 )𝑥𝑥 −(3 𝑥𝑥 6 + 3) 20. (4𝑎𝑎 + 7𝑎𝑎 − 3)𝑎𝑎+ (−8 +𝑎𝑎5 + ) 2 2 3 2 2 3 21. (3𝑝𝑝+ 2− −+ 4𝑝𝑝 )−+ (𝑝𝑝 − 7𝑝𝑝 4 ) 22. ( 𝑏𝑏 5 𝑏𝑏+ 7−) + ( 7 𝑏𝑏 𝑏𝑏) 2 4 3 2 4 4 2 4 23. (8 𝑛𝑛5 +𝑛𝑛5 ) +𝑛𝑛(2− +𝑛𝑛 2− 𝑛𝑛 7 + 1) 𝑛𝑛 − 𝑛𝑛 𝑛𝑛 − 𝑛𝑛 − 𝑛𝑛 4 3 2 2 3 4 24. (6𝑟𝑟 −5 𝑟𝑟 4 𝑟𝑟 ) (2 𝑟𝑟 7 𝑟𝑟 −4 𝑟𝑟 8) (8 6 4 ) 4 2 2 4 2 4 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 − − − 𝑥𝑥 − 𝑥𝑥 Multiply and simplify. Show your work

25. 6( 7) 26. 5 (4 + 4) 4 27. (8𝑝𝑝 −+ 3)(7 5) 28. (3𝑚𝑚 4𝑚𝑚)(5 2) 29. (5𝑏𝑏 + )(6𝑏𝑏 − 4 ) 30. (7𝑣𝑣 +− 5 )(𝑣𝑣8 −+ 3 ) 31. (6𝑥𝑥 𝑦𝑦4)(2𝑥𝑥 − 2𝑦𝑦 + 5) 32. (8𝑥𝑥 + 4𝑦𝑦 +𝑥𝑥6)(6𝑦𝑦 5 + 6) 2 2 2 33. 3(𝑛𝑛3 − 4)(2𝑛𝑛 −+ 1𝑛𝑛) 34. 7(𝑛𝑛 5)(𝑛𝑛 2) 𝑛𝑛 − 𝑛𝑛 35. (6 𝑥𝑥+−3)(6 𝑥𝑥 7 + 4) 36. (5𝑥𝑥 −+ 3 𝑥𝑥+−3)(3 + 3 + 6) 2 2 2 37. (2𝑥𝑥 + 6 +𝑥𝑥 3−)(7𝑥𝑥 6 + 1) 38. 3 𝑘𝑘(6 +𝑘𝑘7) 𝑘𝑘 𝑘𝑘 2 2 2 39. (7𝑎𝑎 + 2𝑎𝑎 3)( 𝑎𝑎 +−4) 𝑎𝑎 40. 3𝑛𝑛 (2𝑛𝑛 + 3)(6 + 9) 2 2 2 𝑢𝑢 𝑢𝑢 − 𝑢𝑢 𝑥𝑥 𝑥𝑥 𝑥𝑥 Find each product by applying the special products formulas. Show your work

41. ( + 8)( 8) 42. (1 + 3 )(1 3 ) 43. (1 7 )(1 + 7 )

44. (𝑥𝑥5 8)𝑥𝑥(5− + 8) 45. (4 + 𝑝𝑝8)(4 − 𝑝𝑝8) 46. (4 − 𝑛𝑛)(4 + 𝑛𝑛) 47. (4𝑛𝑛 − 2 )(𝑛𝑛4 + 2 ) 48. (6𝑥𝑥 2 )(𝑥𝑥6 −+ 2 ) 49. ( 𝑦𝑦+−5𝑥𝑥) 𝑦𝑦 𝑥𝑥 2 𝑚𝑚 − 𝑛𝑛 𝑚𝑚 𝑛𝑛 𝑥𝑥 − 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑎𝑎 237

Chapter 8

50. ( 8) 51. ( + 7) 52. (7 5 ) 2 2 2 53. (𝑥𝑥5 − 3) 54. (𝑝𝑝5 + 7 ) 55. (2 −+ 𝑛𝑛2 ) 2 2 2 56. (5𝑚𝑚+−2 ) 57. (2𝑥𝑥+ 5 𝑦𝑦) 58. (4𝑥𝑥 7𝑦𝑦)(4 + 7) 2 2 59. ( 5𝑟𝑟)( + 5) 60. (4 + 𝑥𝑥2) 61. ( 𝑣𝑣 −4)( +𝑣𝑣4) 2 62. (𝑛𝑛 − 3)(𝑛𝑛 + 3) 63. (8𝑘𝑘 + 5)(8 5) 64. (𝑎𝑎2 −+ 3)𝑎𝑎(2 3) 65. (𝑥𝑥 − 7)(𝑥𝑥 + 7) 66. (7𝑚𝑚+ 7 )(7𝑚𝑚 − 7 ) 67. (3𝑟𝑟 3 )(𝑟𝑟3−+ 3 ) 68. (𝑏𝑏1 +− 5 )𝑏𝑏 69. ( 𝑎𝑎+ 4) 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 70. (1𝑦𝑦 −6 𝑥𝑥) 𝑦𝑦 𝑥𝑥 2 2 2 71. (7 𝑛𝑛7) 72. (𝑣𝑣4 5) 73. (3 −+ 𝑛𝑛3 ) 2 2 2 74. (4𝑘𝑘 − ) 75. (8𝑥𝑥 −+ 5 ) 76. ( 𝑎𝑎 7)𝑏𝑏 2 2 2 77. (8𝑚𝑚+−7𝑛𝑛)(8 7) 78. ( 𝑥𝑥+ 4)(𝑦𝑦 4) 79. (𝑚𝑚7 −+ 7) 2 𝑛𝑛 𝑛𝑛 − 𝑏𝑏 𝑏𝑏 − 𝑥𝑥 Divide: Show your work

80. 81. 82. 4 3 2 4 3 2 4 3 2 20𝑥𝑥 +𝑥𝑥 +2𝑥𝑥 5𝑛𝑛 +𝑛𝑛 +40𝑛𝑛 12𝑥𝑥 +24𝑥𝑥 +3𝑥𝑥 3 4𝑥𝑥 5𝑛𝑛 6𝑥𝑥 83. 84. 85. 5 3 4 2 4 3 2 5𝑥𝑥 +18𝑥𝑥 +4𝑥𝑥 + 9 3𝑘𝑘 +4𝑘𝑘 +2 10𝑛𝑛 +5𝑛𝑛 +2𝑛𝑛 2 2 9𝑥𝑥 8𝑘𝑘 𝑛𝑛 Divide and write your remainder as an expression. Show your work

86. 87. 88. 2 2 2 𝑣𝑣 −2𝑣𝑣−89 𝑥𝑥 −2𝑥𝑥−71 𝑛𝑛 +13𝑛𝑛+32 𝑣𝑣−10 𝑥𝑥+8 𝑛𝑛+5 89. 90. 91. 2 2 2 10𝑥𝑥 −19𝑥𝑥+9 𝑎𝑎 −4𝑎𝑎−38 45𝑝𝑝 −56𝑝𝑝+19

10𝑥𝑥−9 𝑎𝑎−8 9𝑝𝑝−4 92. 93. 94. 2 2 2 27𝑏𝑏 +87𝑏𝑏+35 4𝑟𝑟 −𝑟𝑟−1 𝑛𝑛 −4 3𝑏𝑏+8 4𝑟𝑟+3 𝑛𝑛−2 95. 96. 97. 3 2 3 2 𝑥𝑥 −26𝑥𝑥−41 4𝑥𝑥 −4𝑥𝑥+2 𝑎𝑎 +5𝑎𝑎 −4𝑎𝑎−5 𝑥𝑥+4 2𝑥𝑥−5 𝑎𝑎+7 98. 99. 100. 3 2 3 3 2 𝑝𝑝 +5𝑝𝑝 +3𝑝𝑝−5 𝑥𝑥 −46𝑥𝑥+22 2𝑥𝑥 +12𝑥𝑥 −20

𝑝𝑝+1 𝑥𝑥+7 2𝑥𝑥+6 101. 102. 103. 3 3 2 3 2 4𝑣𝑣 +4𝑣𝑣+19 𝑟𝑟 −𝑟𝑟 −16𝑟𝑟+8 12𝑛𝑛 +12𝑛𝑛 −15𝑛𝑛−4 4𝑣𝑣+12 𝑟𝑟−4 2𝑛𝑛+3

238

Chapter 8 CHAPTER REVIEW KEY TERMS AND CONCEPTS Look for the following terms and concepts as you work through the workbook. In the space below, explain the meaning of each of these concepts and terms in your own words. Provide examples that are not identical to those in the text or in the media lesson.

Product rule of exponents

Quotient rule of exponents

Power rule of a product

Power rule of a quotient

Power rule of a Power

Zero power rule

Negative exponent rule

Reciprocal of negative rule

Negative power of a quotient rule

Scientific notation

Standard notation

(Decimal notation)

Polynomial

Monomial

239

Chapter 8

Binomial

Trinomial

Leading Term

Leading Coefficient

Degree of a Polynomial

Constant Term

240