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Algebraic Expressions What are the Terms? 2x + 3y - 7 Algebraic Expressions Terms 2x + 3y - 7 Algebraic Expressions What are the variables? 2x + 3y - 7 Algebraic Expressions Variables 2x + 3y - 7 Algebraic Expressions What are the coefficients? 2x + 3y - 7 Algebraic Expressions Coefficients 2x + 3y - 7 Algebraic Expressions What is the constant? 2x + 3y - 7 Algebraic Expressions 2x + 3y - 7 Constant Algebraic Expressions Polynomial: monomial → x, 2xy, 4, 3x²y, … single term binomial → x+1, 2xy+x, 3x²y+4, …two terms trinomial → 2x+3y+7, 3x²y+xy+4x, …three terms polynomial → …four or more terms What is the area of a rectangle? Length times Width If the length is 3 meters and the width is 2 meters, what is the area? A=LxW A = 3 x 2 = 6 meters2 A, L and W are the variables. It is any letter that represents an unknown number. An algebraic expression contains: 1) one or more numbers or variables, and 2) one or more arithmetic operations. Examples: x-3 3 • 2n 4 1 m In expressions, there are many different ways to write multiplication. 1) 2) 3) 4) 5) ab a•b a(b) or (a)b (a)(b) axb We are not going to use the multiplication symbol any more. Why? Division, on the other hand, is written as: x 1) 3 2) x ÷ 3 Here are some phrases you may have see throughout the year. The terms with * are ones that are often used. Addition Subtraction Multiplication Division sum* difference* product* quotient* increase decrease times divided plus minus multiplied ratio add subtract more than less than total Write an algebraic expression for 1) m increased by 5. m+5 2) 7 times the product of x and t. 7xt or 7(x)(t) or 7 • x • t 3) 11 less than 4 times a number. 4n - 11 4) two more than 6 times a number. 6n + 2 5) the quotient of a number and 12. x 12 Which of the following expressions represents 7 times a number decreased by 13? 1. 2. 3. 4. 7x + 13 7x - 13 13 - 7x 13 + 7x Which one of the following expressions represents 28 less than three times a number? 1. 2. 3. 4. 28 - 3x 3x - 28 28 + 3x 3x + 28 Write a verbal expression for: 1) 8 + a. The sum of 8 and a 2) m r . The ratio of m to r Do you have a different way of writing these? Which of the following verbal expressions represents 2x + 9? 1. 2. 3. 4. 9 increased by twice a number a number increased by nine twice a number decreased by 9 9 less than twice a number Which of the following expressions represents the sum of 16 and five times a number? 1. 2. 3. 4. 5x - 16 16x + 5 16 + 5x 16 - 5x Which of the following verbal expressions represents x2 + 2x? 1. 2. 3. 4. the sum of a number squared and twice the number the sum of a number and twice the number twice a number less than the number squared the sum of a number and twice the number squared Which of the following expressions represents four less than the cube of a number? 1. 2. 3. 4. 4 – x3 4 – 3x 3x – 4 x3 – 4 Evaluate. 21 22 23 2n7 2 2•2=4 2•2•2=8 We can’t evaluate because we don’t know what n equals to!! Competition Problems Evaluating Algebraic Expressions Evaluate the following algebraic expression using m=7, n=8 n² - m Answer: 57 Evaluate the following algebraic expression using x=5, y=2 8(x-y) Answer: 24 Evaluate the following algebraic expression using x=7, y=2 yx ÷ 2 Answer: 7 Evaluate the following algebraic expression using x=1, z=19 z + x³ Answer: 20 Evaluate the following algebraic expression using m=3, p=10 15-(m+p) Answer: 2 Evaluate the following algebraic expression using a=9, b=4 b(a+b) + a Answer: 61 Evaluate the following algebraic expression using m=3, p=4 p²÷4-m Answer: 1 Evaluate the following algebraic expression using x=4, y=2 y(x-(9-4y)) Answer: 6 Evaluate the following algebraic expression using x=9, y=1 x-(x-(x-y³)) Answer: 8 Evaluate the following algebraic expression using h=9, j=8 j(h-9)³ +2 Answer: 2 Simplifying Algebraic Expressions REVIEW Insert Lesson Title Here Vocabulary term coefficient like terms The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms. Like terms 4x – 3x + 2 Constant A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1. Coefficients 1x2 + 3x In the expression 7x + 5, 7x and 5 are called terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by + and –. 7x term + 5 term – 3y2 + term y + x 3 term term In the term 7x, 7 is called the Coefficient coefficient. A coefficient is a number that is multiplied by a variable in an algebraic expression. A variable by itself, like y, has a coefficient of 1. So y = 1y. Variable Term Coefficient 4a 2 3 3k2 x2 x 9 4.7t 4 2 3 3 1 1 9 4.7 Like terms are terms with the same variable raised to the same power. The coefficients do not have to be the same. Constants, like 5, 12 , and 3.2, are also like terms. Like Terms Unlike Terms 3x and 2x 5x2 and 2x The exponents are different. w and w 7 6a and 6b The variables are different 5 and 1.8 3.2 and n Only one term contains a variable Additional Example 1: Identifying Like Terms Identify like terms in the list. 3t 5w2 7t 9v 4w2 8v Look for like variables with like powers. 3t 5w2 7t 9v 4w2 8v Like terms: 3t and 7t, 5w2 and 4w2, 9v and 8v Helpful Hint Use different shapes or colors to indicate sets of like terms. Insert Lesson Title Here Identify like terms in the list. 2x 4y3 8x 5z 5y3 8z Look for like variables with like powers. 2x 4y3 8x 5z 5y3 8z Like terms: 2x and 8x, 4y3 and 5y3 , 5z and 8z Insert Lesson Title Here Combining like terms is like grouping similar objects. x x x + x 4x x x + x x 5x x = = x x x x x x 9x To combine like terms that have variables, add or subtract the coefficients. x x x Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression. 7x2 – 4x2 = (7 – 4)x2 = (3)x2 Factor out x2 from both terms. Perform operations in parenthesis. = 3x2 Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same. Simplify the expression by combining like terms. 72p – 25p 72p – 25p 47p 72p and 25p are like terms. Subtract the coefficients. Simplify the expression by combining like terms. A variable without a coefficient has a coefficient of 1. and Write 1 as are like terms. . Add the coefficients. Simplify the expression by combining like terms. 0.5m + 2.5n 0.5m + 2.5n 0.5m and 2.5n are not like terms. 0.5m + 2.5n Do not combine the terms. Simplify by combining like terms. 16p + 84p 16p + 84p 100p 16p + 84p are like terms. Add the coefficients. –20t – 8.5t2 –20t – 8.5t2 20t and 8.5t2 are not like terms. –20t – 8.5t2 Do not combine the terms. 3m2 + m3 3m2 + m3 3m2 and m3 are not like terms. 3m2 + m3 Do not combine the terms. Simplify 14x + 4(2 + x) Procedure 1. 2. 14x + 4(2 + x) 14x + 4(2) + 4(x) 3. 14x + 8 + 4x 4. 14x + 4x + 8 5. (14x + 4x) + 8 6. Justification 18x + 8 Distributive Property Multiply. Commutative Property Associative Property Combine like terms. Simplify 6(x – 4) + 9. Justify each step. Procedure 1. 6(x – 4) + 9 2. 6(x) – 6(4) + 9 3. 6x – 24 + 9 4. 6x – 15 Justification Distributive Property Multiply. Combine like terms. Simplify −12x – 5x + 3a + x. Justify each step. Procedure 1. –12x – 5x + 3a + x 2. –12x – 5x + x + 3a 3. –16x + 3a Justification Commutative Property Combine like terms. Simplify each expression. 165 +27 + 3 + 5 200 8 Write each product using the Distributive Property. Then simplify. 5($1.99) 6(13) 5($2) – 5($0.01) = $9.95 6(10) + 6(3) = 78 Simplify each expression by combining like terms. Justify each step with an operation or property. 14c2 – 9c 14c2 – 9c 301x – x 300x 24a + b2 + 3a + 2b2 27a + 3b2 Let’s work more problems… Simplify the following algebraic expression: -3p + 6p Answer: 3p Simplify the following algebraic expression: 7x - x Answer: 6x Simplify the following algebraic expression: -10v + 6v Answer: -4v Simplify the following algebraic expression: 5n + 9n Answer: 14n Simplify the following algebraic expression: b - 3 + 6 - 2b Answer: -b + 3 Simplify the following algebraic expression: 10x + 36 - 38x - 47 Answer: -28x - 11 Simplify the following algebraic expression: 10x-w+4y-3x+36-38x-47+32x+2w-3y Answer: w+x+y-11 Simplify the following algebraic expression using the distributive property: 6(1 – 5m) Answer: 6 – 30m Simplify the following algebraic expression using the distributive property: -2(1 – 5v) Answer: -2 + 10v Simplify the following algebraic expression using the distributive property: -3(7n + 1) Answer: -21n - 3 Simplify the following algebraic expression using the distributive property: (x + 1) ∙ 14 Answer: 14x + 14 Simplify the following algebraic expression using the distributive property: (3 - 7k) ∙ (-2) Answer: -6 + 14k Simplify the following algebraic expression using the distributive property: -20(8x + 20) Answer: -160x - 400 Simplify the following algebraic expression using the distributive property: (7 + 19b) ∙ -15 Answer: -105 – 285b Variable Expressions x 5 means (use parentheses for multiplica tion) ( x)( x)( x)( x)( x) y 3 means ( y)( y )( y) Simplify: (-a)² Answer: a² Substitution and Evaluating STEPS 1. Write out the original problem. 2. Show the substitution with parentheses. 3. Work out the problem. Example : Solve if x 4; Solve if x 4; (4) 3 x3 = 64 x3 Evaluate the variable expression when x = 1, y = 2, and w = -3 ( x) ( y ) 2 2 ( x y) Step 1 ( x) ( y ) 2 2 ( x y) Step 1 2 wx y Step 2 Step 2 (1) (2) 2 Step 3 1 4 5 wx y Step 1 Step 2 (1) 2 (2) 2 2 Step 3 (3) 9 2 (3)(1) 2 Step 3 (3)(1) 3 Contest Problem Are you ready? 3, 2, 1…lets go! Evaluate the expression when a= -2 a² + 2a - 6 Answer: -6 Evaluate the expression when x= -4 and t=2 x²(x-t) Answer: -96 Evaluate the expression when y= -3 (2y + 5)² Answer: 1 MULTIPLICATION PROPERTIES PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base. 2 2 3 3 5 4 ( x )( x ) (2 2 2)(2 2 2 2 2) ( x)( x)( x) ( x)( x)( x)( x) 28 x7 256 MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions. (6 5) 2 (5 xy) 3 (6) 2 (52 ) 3 3 3 (5 )( x )( y ) (4 x 2 ) 3 x 5 (64)( x 6 ) x 5 (43 )( x 2 )3 x 5 64x11 36 25 900 125 x 3 y 3 (64) ( x 2 )( x 2 )( x 2 ) x 5 Problems Are you ready? 3, 2, 1…lets go! Simplify. Your answer should contain only positive exponents. 2n⁴ · 5n ⁴ Answer: 10n⁸ Simplify. Your answer should contain only positive exponents. 6r · 5r² Answer: 30r³ Simplify. Your answer should contain only positive exponents. 6x · 2x² Answer: 12x³ Simplify. Your answer should contain only positive exponents. 6x² · 6x³y⁴ Answer: 36x⁵y⁴ Simplify. Your answer should contain only positive exponents. 10xy³ · 8x⁵y³ Answer: 80x⁶y⁶ MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base. 2 3 (5 ) (x 2 ) 4 (52 )(52 )(52 ) ( x 2 )( x 2 )( x 2 )( x 2 ) 56 x8 MULTIPLICATION PROPERTIES SUMMARY PRODUCT OF POWERS x x x a b a b ADD THE EXPONENTS POWER TO A POWER x a b x a b MULTIPLY THE EXPONENTS POWER OF PRODUCT ( xy) x y a a a Problems Are you ready? 3, 2, 1…lets go! Simplify. Your answer should contain only positive exponents. (a²)³ Answer: a⁶ Simplify. Your answer should contain only positive exponents. (3a²)³ Answer: 27a⁶ Simplify. Your answer should contain only positive exponents. (x⁴y⁴)³ Answer: x¹²y¹² Simplify. Your answer should contain only positive exponents. (2x⁴y⁴)³ Answer: 8x¹²y¹² Simplify. Your answer should contain only positive exponents. (4x⁴·x⁴)³ Answer: 64x²⁴ Simplify. Your answer should contain only positive exponents. (4n⁴·n)² Answer: 16n¹⁰ ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1. 33 27 32 9 31 3 30 1 1 1 3 1 3 3 1 1 2 3 2 3 9 1 1 3 3 3 3 27 1 1 2 2 x 2 2 2 x x 1 1 1 2 (2 x) 2 2 2 2 (2 x) 2 x 4x 2 1 1 1 34 4 1 1 3 81 4 4 1 3 3 1 34 DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base. x5 ( x)( x)( x)( x)( x) ( x)( x) 2 x 3 x ( x)( x)( x) 1 1 x 3 x 3 1 1 1 1 4 4 3 x 3 4 7 4 x x x x x x DIVISION PROPERTIES POWER OF A QUOTIENT 4 x ( x 2 )4 x8 3 3 4 12 (y ) y y 2 Hard Example 3 3 3 6 2 3 3 12 2 xy 2 x y ( 2 xy ) 8 x y 3 4 3 9 12 3 4 3 9 6 3 x y (3x y ) 27 x y 3x y 2 8 x 3 y12 8 y6 9 6 27 x 6 27 x y ZERO, NEGATIVE, AND DIVISION PROPERTIES Zero power ( x) 1 Negative Exponents x a 1 a x and 1 a x a x 0 Quotient of powers xa a b x b x Power of a quotient a x x a y y a Problems Are you ready? 3, 2, 1…lets go! Simplify. Your answer should contain only positive exponents. 3r³ 2r Answer: 3r² 2 Simplify. Your answer should contain only positive exponents. 3xy 5x² 2 ( ) Answer: 9y² 25x² Simplify. Your answer should contain only positive exponents. 18x⁸y⁸ 10x³ Answer: 9x⁵y⁸ 5 Simplify: (x⁴y¯²)(x¯¹y⁵) Answer: x³y³ Simplify the following algebraic expression using the distributive property: 8x ∙ (6x + 6) Answer: 48x² + 48x Simplify the following algebraic expression using the distributive property: 7n(6n + 3) Answer: 42n² + 21n Simplify the following algebraic expression using the distributive property: 2(9x – 2y) Answer: 18x – 4y Simplify the following algebraic expression using the distributive property: 1 + 7(1 – 3b) Answer: 8 - 21b Simplify the following algebraic expression using the distributive property: -2 - 7(-1 – 3b) Answer: 5 + 21b Simplify the following algebraic expression using the distributive property: 3n(n² - 6n + 5) Answer: 3n³ - 18n² + 15n Simplify the following algebraic expression using the distributive property: 2k³(2k² + 5k - 4) Answer: 4k⁵ +10k⁴ - 8k³ Simplify the following algebraic expression using the distributive property: 9(x² + xy – 8y²) Answer: 9x² + 9xy – 72y² Simplify the following algebraic expression using the distributive property: 9v²(u² + uv - 5v²) Answer: 9v²u² +9v³u – 45v⁴ Simplify the following algebraic expression using the distributive property: 3x(5x+2) - 14(2x²-x+1) Answer: -13x² + 20x - 14 Simplify completely: 4x²y 2x Answer: 2xy Simplify completely: y¯¹ y¯² Answer: y Simplify completely: 16x⁴y¯¹ 4x²y¯² Answer: 4x²y Simplify completely: 36x³y⁶z¹² 4x¯¹y³z¹⁰ Answer: 9x⁴y³z² Simplify completely: 21x³y⁷z¹⁴ 18x⁴y⁶ · 30x³z¯⁵ y¹²z¯⁶ Answer: 35x²z¹⁵ y¹¹