Algebra I. Exponents and Polynomials. Name
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1 Algebra I Exponents and Polynomials Name 1
2 2
3 UNIT SELF-TEST QUESTIONS The Unit Organizer #6 2 LAST UNIT /Experience NAME 4 BIGGER PICTURE DATE Operations with Numbers and Variables 1 CURRENT CURRENT UNIT UNIT 3 Quadratics NEXT UNIT /Experience 8 UNIT SCHEDULE 5 UNIT MAP UO Launch expressions using the laws of exponents applying rules and processes to A.11(B) problems by combining or multiplying polynomials A.10(A,B) trinomials into 2 binomials A.10(E) Unit 6 Test Words to know: Polynomial Term Exponent Like Term Factor FOIL Degree Combine Binomial Trinomial Radical Radicand Perfect Square 6 7 RELATIONSHIPS UNIT 3
4 Key Words and Processes Word or Process Definition/Example/Illustration Factor Base Exponent Product Rule for Exponents (multiplying) 4
5 Key Words and Processes Word or Process Definition/Example/Illustration Quotient Rule for Exponents (dividing) Power Rule for Exponents (exponent to a power) Negative Exponent Rule Zero Exponent Rule 5
6 Key Words and Processes Word or Process Definition/Example/Illustration The like exponent rule The Square Root Exponent Rule 6
7 Key Words and Processes Word or Process Definition/Example/Illustration Term Like Term Coefficient Polynomial 7
8 Key Words and Processes Word or Process Definition/Example/Illustration Variable Constant Operator Degree of a Polynomial 8
9 Key Words and Processes Word or Process Definition/Example/Illustration Leading Coefficient Combine Like Terms Binomial Trinomial 9
10 Key Words and Processes Word or Process Definition/Example/Illustration FOIL Method for multiplying two binomials Box Method for multiplying two polynomials Factoring The sum of the factors method for factoring a trinomial 10
11 Key Words and Processes Word or Process Definition/Example/Illustration Graphing Method for factoring a trinomial Square Root Radical and Radicand The like Exponent Rule 11
12 Key Words and Processes Word or Process Definition/Example/Illustration GCF (greatest common factor) Perfect Square Simplifying a Radical Expression Combining Radical Expressions 12
13 Exponents is about: Base Exponent (power) Base & Exponent Expanded Form Value Ratio of Expanded Forms , , =
14 Multiplication with Exponents Base & Exponent Expanded Form Value In Exponent Form (10)( ) 1,000, Division with Exponents Base & Exponent Expanded Form Value In Exponent Form = 10*10*10*10 10, What Patterns do we notice? So What? 14
15 Operations with Exponents is about: So What? Remember: 1. When no exponent is shown the exponent = 1 2. Anything to the zero power (Except zero) =1 15
16 Using the Laws of Exponents When multiplying if the bases are the same we the exponents Examples: = 3 5 x 3 x 2 = x 5 x 4 x 5 = x = 3 3 x 5 x 2 = x 3 If there are coefficients in front of the exponents we them Examples: Practice: 2x 3 4x 2 = 8x 5 3x 4 6x 5 = 18x 9 1. x 5 x 2 = 2. x 4 x = 3. x 8 x 5 = 4. 2x 4 3x 2 = 5. 4x 3 2x 2 = 16
17 When dividing if the bases are the same we take the of the exponents Examples: = 3 1 x 3 x 2 = x x 4 x 5 = x 1 x 5 x 2 = x 7 If there are coefficients in front of the exponents we can them Examples: 9x 3 3x 2 = 3x 1 3x 5 6x 3 = 1 2 x2 We usually write these problems as fractions instead of using the division symbol. Practice: 1. x5 x 2 = 2. x7 x 3 = 3. 8x6 2x 1 = 4. 3x5 6x 4 = 17
18 When taking an exponent to an exponent we the exponents Examples: (x 3 ) 5 = x 15 (x 4 ) 2 = x 8 (x 2 ) 3 = x 6 (x) 5 = x 5 Practice: 1. (x 2 ) 3 = 2. (x 3 ) 4 = 3. (x 4 ) 2 = 4. (x 3 ) 5 = 5. (x 2 ) 2 = 18
19 We can use the Laws of Exponents to solve for missing values Examples: x 7 x? = x 10 The missing? is 3 since 10-7 = 3. The missing? is 5 since 8 3 = 5 x 8 x? = x3 (x 3 )? = x 12 The missing? is 4 since 12/3 = 4 If = x 3, then y 5 = x 15 Practice Find the value for? 1. x 5 x? = x x9 x? = x5 3. (x 2 )? = x If y = x 2, then y? = x 8 19
20 When not all the bases are the same we can still use the Laws of Exponents on the parts of the problem that have the same base to simplify the expression since multiplication is communitive. Examples: 3x 5 y 3 z 2 4x 3 y 2 = 12x 8 y 5 z 2 15x 4 y 5 z 2 3x 2 y 2 z = 5x2 y 3 z 20
21 Practice Quiz: 21
22 22
23 Use the Laws of Exponents to find the missing value ( a ) Multiplication 1. x 8 x a = x x 3 x a = x 1 3. x 20 x a = x 5 4. x 12 x a = x 15 Division 1. x 6 = x2 xa 2. x 3 = x2 xa 3. x 8 = x2 xa 4. x 9 = x2 xa 23
24 Exponent to Exponent 1. (x 4 ) a = x (x 10 ) a = x 5 3. (x 10 ) a = x (x 15 ) a = x 30 Combining Rules 1. ( x8 x a)4 = x ( x3 x a)9 = x ( x3 x a)8 = x 4 4. ( x7 x a)4 = x 4 24
25 25
26 Exponent Problems 1. The length of a rectangular plot of land is given by the expression 6x 3 y 4 and the width is given by the expression 5x 5 y 4 z. Write the expression that shows the area of the plot. 2. The area of a rectangle is given by the expression 36x 6 y 2. If the length is 4x 2 y 1 what expression shows the width? 3. A cube has a side length of 2x 3. What expression shows the volume of the cube? 4. Find the expression that gives the volume of this rectangular prism
27 6. The area of a rectangle is given by the equation 2x 2 = 800, where x is the width of the rectangle. What is the length of the rectangle? 7. If y = x 5 then rewrite the equation without the negative exponent and evaluate it for when x = 2 8. If x = n 3 write the equation in terms of x that is equivalent to n What is the value of y if y = x5 when x = 3? 2 x 10. What is the value of y if y = x3 when x = 2? 6 x 11. The radius, r, of a tree is 3x and the height, h, is 12x, where x is the age of the tree in years. If we assume the tree is a cylinder and the volume of a cylinder is given by the formula V = πr 2 h, write the expression for the volume of the tree in terms of x. 12. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. If r is given by the expression 4x write the expression for the circumference of the circle in terms of x. The Open Box Problem 27
28 28
29 Polynomials The FRAME Routine Key Topic Polynomials is about Unit 6 Main idea Traits Essential details Main idea Classify Main idea Operations Essential details So What? (What s important to understand about this?) 29
30 Identifying Polynomials Term Polynomial Yes or No? If no, why? If yes, Type If yes, Degree If yes, Leading Coefficient 30
31 Like Term Review Expression Like Term or Not? Justification 4x and 3 4x and 3y 4x and 3x 2 4x and 3x 4 and 3 2x and 2y If terms have the same and they are called and we can combine (add/subtract) the coefficients 31
32 Naming Polynomials Practice 32
33 Simplify each polynomial expression by combining like terms 33
34 Multiplying Polynomials is about Using: 1. The Laws of Exponents 2. The distributive property 3. Combining Like Terms 4. Using Different Models/Methods 34
35 35
36 So What? 36
37 37
38 Practice Using the Rectangular Method 38
39 Practice Using the Distributive Model 39
40 Multiplying Polynomials Practice: Multiply each set of polynomials and simplify the produce by combining like terms, if there are like terms. 40
41 41
42 Multiplying Binomials: The most important polynomials we will study in Algebra I are binomials and trinomials and how to convert between the two forms by combining like terms or later in the unit by factoring. Trinomials are in the form: ax 2 bx c Where a and b are coefficients and c is a constant. Find the product of each set of binomials and then simplify by combining like terms. 42
43 Applications of Multiplying Polynomials 1. Find the area of the triangle shown below. 2. The perimeter of the triangle shown below is 13x 2 14 x 8. What expression shows the length of the side not shown? 3. What is the ratio of the area of the square to the area of the circle if r is the radius of the circle? 43
44 4. A rectangle has a length of 4m 3 n 2 and a width of 6m 3 n 2. Write the expression for the area of this rectangle. 5. One base of a trapezoid is (x + 2) centimeters and the other base is (x + 4) centimeters. The height of the trapezoid is 4 centimeters. a. Express the area of the trapezoid in terms of x b. What is the area if x = 3? 6. A right triangle has a base of (x 3) inches and a height of (x + 4) inches. If the perimeter of the triangle is (3x + 7) inches. a. What is the length of the hypotenuse in terms of x? b. What is the area of the triangle in terms of x? c. What is the perimeter and area if x = 11? 7. A rectangle has a width of (2x 5) centimeters and a length of x 2 2x 3 centimeters. a. What is the perimeter of the rectangle in terms of x? b. What is the area of the rectangle in terms of x? 44
45 8. The length, L. of a rectangle is 3 feet more than its width, W. a. Draw and label a diagram to match this situation. b. Express the perimeter in terms of W. c. Express the area in terms of W. d. What is the perimeter if W = 7? 45
46 Factoring In the same way that we can multiply two binomials to create a trinomial, after we combine like terms we can reverse the process to factor a trinomial into 2 binomials. Factors of Whole Numbers: 1) What are the factors of 48? 2) What are the factors of 60? 3) What are the factors of 16? 4) What are the factors of 36? 5) What are the factors of 25? In a similar way that we can factor whole numbers we can factor many polynomials. The image we can use is finding the area of rectangle. The area can be expressed as the trinomial and the length and width are the two binomials. What is the area of the rectangles below, expressed as a trinomial in the form ax 2 bx c (x+3) (x+2) 46
47 (x+7) (x+4) (x - 5) (x + 2) (x - 4) (x - 3) What do you notice about a, b, and c in the trinomials and how they relate to the constants in the binomials? 47
48 Factoring Practice In your groups, create 2 examples of trinomials that you can factor into 2 binomials and be ready to share these with the class. 48
49 Group Practice 49
50 50
51 Factoring Practice 51
52 Factoring Special Cases 52
53 We can also factor just by find the GCF (greatest common factor) by applying the laws of exponents combined with rules for GCF To factor the polynomial 6x 4 12x 3 + 4x 2, for example, follow these steps: 1. Break down every term into prime factors. This step expands the original expression to 2. Look for factors that appear in every single term to determine the GCF. In this example, you can see one 2 and two x s in every term: The GCF here is 2x Factor the GCF out from every term in front of parentheses and group the remnants inside the parentheses. You now have 4. Multiply each term to simplify. The simplified form of the expression you find in Step 3 is 2x 2 (3x 2 6x + 2). To see if you factored correctly, distribute the GCF and see if you obtain your original polynomial. If you multiply the 2x 2 inside the parentheses, you get 6x 4 12x 3 + 4x 2. You can now say with confidence that 2x 2 is the GCF. 53
54 Find the GCF of each pair of monomials 54
55 55
56 Factor each of these polynomials: Do problems #1, #3, #5, #7, #8, #9, #10, #12, #15, #17 When and how can we use the graphing calculator to factor? 56
57 57
58 Simplifying Radicals (square roots) 58
59 Simplifying Radicals is about.. Example 3: Simplify 75 So What? 59
60 Why does this work? The like exponent rule a y a = y a Examples First Term Second Term Equals Simplified = 9 = 16 9 * 16 = 144 ( ) = = 5 = 25 = 4 25 * 4 = 100 (5 ) = 0 = 00 6 = 6 6 = 6 * 4 = = 5 6 = = = 3 * 7 = 21 (9 9) 0.5 = 0.5 = Note: Taking the square root of a number is the same as raising it to the 0.5 or power. Try some on the calculator. 5 = = =
61 Fill on the table below to familiarize yourself with the perfect squares Practice: Proof: Use the calculator to show that your simplified expression equals the original radical by using the square root function. 61
62 Additional Practice 62
63 Simplifying Radicals with Variables in the Radicand 63
64 Practice: 64
65 65
66 66
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