Chapter 5: Exponents and Polynomials

Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5 Multiplication with Polynomials 5.6 Binomial Squares and other Special Products 5.7 Division with Polynomials 1

5.1 What is an Exponent? Consider Exercise: Write each expression as a single number. 2

Multiplication with Exponents Consider Method 1: (Order of Operations) Method 2: (Using Exponents) The Product Rule 3

Ex: Simplify 1) 4) 7) (w 3 z 4 )(w 3 z 7 ) 2) 5) 3) 6) (x + y) 6 (x + y) 9 4

Consider Method 1: (Order of Operations) Method 2: (Using Exponents) The Power Rule 5

Ex: Simplify 1) 3) 2) 4) [(x+y) 3 ] 27 6

Consider Method 1: (Order of Operations) Method 2: (Using Exponents) Raising a product to a power 7

Ex: Simplify 1) 3) 2) 8

Scientific Notation Definition: A number is in scientific notation when it is written as the product of a number between 1 and 10 and an integer power of 10. A number written in scientific notation takes the form: where and r is an integer. Examples Which are in scientific notation and which are not? 9

5.2 Division with Exponents Consider: Negative Exponents What is, for example? Using our product rule, we know that so, let's solve for in 11

Negative Exponent Rule and Examples Write each expression with a positive exponent. 12

Division with Exponents Consider Method 1: (Order of Operations) Method 2: (Using Exponents) The Quotient Rule 13

Ex: Simplify 1) 3) 5) (x+y) 9 (x+y) 2 2) 4) 14

Consider Raising a quotient to a power 15

Ex: Simplify 1) 3) 2) 16

Consider What should equal? By quotient rule, The Exponent Zero 17

Ex: Simplify 1) 3) 2) 4) 18

Application Suppose you have two squares, one of which is larger than the other. If the length of a side of the larger square is 3 times as long as the length of a side of the smaller square, how many of the smaller squares will it take to cover up the larger square? 19

Application Suppose you have two boxes, each of which is a cube. If the length of a side of the larger box is 3 times as long as the length of a side of the smaller box, how many of the smaller boxes will fit inside the larger box? 20

Scientific Notation with negative exponents Definition: A number in scientific notation with a negative exponent indicates a very small number. A number in scientific notation with a large positive integer indicates a large number. where and r is a negative integer. Examples Convert from scientific notation to decimal form. 21

Definitions & Property of Exponents Overview 23

5.3 Operations with Monomials Definition: A monomial is a one-term expression that is either a constant (number) or the product of a constant and one or more variables raised to whole number exponents. Examples 24

5.3 Operations with Monomials Definition: the degree of a monomial in one-variable is the exponent on the variable. If a monomial has multiple variables, the degree is the sum of all the exponents. Examples 25

5.3 Operations with Monomials Multiplying Monomials. (Which rule of exponents are you using?) Examples 26

5.3 Operations with Monomials Dividing Monomials. (Which rule of exponents are you using?) Examples Example Divide by 27

5.3 Operations with Monomials Examples Simplify 28

Multiplication and Division of Numbers written in Scientific Notation Examples Multiply Examples Divide Examples Simplify 29

Addition and Subtraction of Monomials Definition Two terms (monomials) with the same variable part (same variables raised to the same powers) are called similar or like terms. Examples Combine the like terms 30

Addition and Subtraction of Monomials Examples Simplify 31

Addition and Subtraction of Monomials Application A rectangular solid is twice as long as it is wide and one-half as high as it is wide. Write an expression for the volume. 32

5.4 Addition and Subtraction of Polynomials Definition A polynomial is a finite sum of monomials (terms). Definition The degree of a polynomial is the degree of the leading term once the polynomial is written in standard form. (highest to lowest power) 33

5.4 Addition and Subtraction of Polynomials Examples: 1. Write the polynomial in standard form: 2. Find the value of when 3. Add: 4. Add and 34

5.4 Addition and Subtraction of Polynomials Examples: 5. Find the opposite polynomial to 6. Subtract: 7. Subtract from 35

Rule: Case 1: Product of Monomials 1) (5x)(3x) 36

Rule: 2) x(x - 5) 37

Rule: 2) x(x - 5) 3) 4x(x - 5) 4) -4x(x - 5) 38

Note the differences between addition/subtraction vs. multiplication 39

Consider (x + 3)(x + 2). Use the box method: Use the distributive property: 40

Consider (x + 3)(x + 2). 41

1) (x - 3)(x + 2) 2) (x - 5)(x - 7) 3) (3x - 4)(2x + 5) 4) (4x - 1) (6x + 1) 42

Quick Check! (x - 5) + (x + 2) (x - 5)(x + 2) 43

Find the area & perimeter of the rectangle. 46

A. The Square of a Binomial. When both binomials are the same. Ex: Multiply and explore these products: Do you see a pattern? 47

Perfect Square Trinomials The Box Area Method The FOIL Method 48

Perfect Square Trinomials Ex: Identify A and B for each of the following: 49

Perfect Square Trinomials Ex: Use the formula to JUMP and write the result, 50

Mixed Exercises Ex: Simplify: 51

B. The Product of a Sum and a Difference when you have the SAME Two Numbers. Ex: Multiply and explore the products: Do you see a pattern? 52

Difference of two squares pattern (A + B)(A B) = A 2 B 2 The Box Area Method The FOIL Method 53

Difference of two squares pattern (A + B)(A B) = A 2 B 2 Ex: Identify the 'A' and 'B' in the following products. 54

Difference of two squares pattern (A + B)(A B) = A 2 B 2 Ex: Use the formula to JUMP and write the result,. 55