1. Unit 2: Expressions Section 1: Algebraic Expressions Numerical expressions are those which contain only numbers and operation symbols Algebraic expressions are those that contain one or more variables, in addition to numbers and operation symbols Expressions are not sentences because they do not contain any verbs, such as equal signs or inequality signs Finding the numerical value using the order of operations is known as evaluating an expression

2. Order of operations: – 1) Parentheses: perform all operations within the parentheses, following the order of operations – 2) Exponents: perform all operations containing exponents (if you have a negative number raised to a power, you must put the number in parentheses with the exponent outside when using a calculator) – 3) Multiplication and division: go from left to right to determine the order – 4) Addition and subtraction: go from left to right to determine the order When evaluating an expression, you must show work

3. Ex1. Evaluate when x = 6 and y = 5 Ex2. Evaluate Ex3. Evaluate when x = -3 Sections from the book to read: 1-1 and 1-4

4. Section 2: Understanding Terms Individual expressions are also called terms Terms have no operation symbols or verbs If a term contains a number, that number is called the coefficient (i.e. with the term 6x the 6 is the coefficient) Terms can have multiple variables (i.e. 8ab³cd² is a single term) In an expression, one or more terms can be combined by addition or subtraction (i.e. 7x + 2y is two terms added together)

5. An expression with one term is a monomial An expression with two terms is a binomial An expression with three terms is a trinomial An algebraic expression that is either a monomial or a sum of monomials is a polynomial Each individual term has a degree The degree of a monomial is the sum of the exponents of each of the variables Ex1. Find the degree of each monomial A) B)

6. To find the degree of a polynomial, find the degree of each term. The highest number is the degree of the entire polynomial (or expression) Ex3. Find the degree of each expression. A) B) If an expression has a degree of 1 then it is linear (the graph would be a line) If an expression has a degree of 2 then it is a quadratic (the graph would be a parabola) When an expression is in standard form, the terms are in descending order of the exponents of its terms

7. Ex4. Name the type of expression (monomial, binomial, trinomial, other) a) 5x + 3y – 2 b) 9a + 12 c) 8c + 9d + 2e – 5f d) 3xyz Sections from the book to read: 3-6 and 10-1

8. Section 3: Adding Like Terms Like terms are those with EXACTLY matching variables You can add the coefficients to like terms because of the distributive property (used in reverse) Distributive property: a(b + c) = ab + ac (b + c)a = ab + ac For example: 3x + 5x = (3 + 5)x = 8x You cannot add terms that are unlike Ex1. Simplify: 3a + 2b + 8a + b

9. Ex2. Simplify Notice that the exponents do not change when you add (or subtract) like terms Ex3. Simplify Ex4. Simplify -3x + 2y + 8x Section from the book to read: 3-6

10. Section 4: Subtracting Like Terms When you are subtracting like terms, it may be beneficial to change subtraction to adding the opposite (this is your choice) Just like with addition, you can only subtract LIKE TERMS Use the distributive property to simplify and write the answer in standard form Ex1. 10x – 8y – 4x – (-2y) Ex2. 3m² + 8m + (-12m) – 7m² – 9m

11. If there is a negative sign or a subtraction sign directly outside of a set of parentheses containing either a sum or a difference, you distribute the sign to each term within the parentheses Opposite of a Sum Property: For all real numbers a and b, -(a + b) = -a + -b = -a – b Opposite of Opposite Property (Op-op prop): For an real number a, -(-a) = a Opposite of a Difference Property: For all real numbers a and b, -(a – b) = -a + b

12. Simplify each expression. Ex3. 10x – (5x + 8) + 12 – 3x Ex4. (5n – 8p) – (9n – 5p) + 4p Ex5. -8y – (7y – 4z + 2) + 6z Ex6. Ex7. Sections from the book to read: 4-5

13. Section 5: Chunking Chunking is a technique of grouping repeated expressions together in order to simplify in an easier way For example: 3(2x + 6) + 8(2x + 6). The “chunk” would be 2x + 6 because it is repeated. Think of 2x + 6 like a single variable, y. 3y + 8y = 11y so it is like 11(2x + 6). Distribute in the last step to get 22x + 66 This technique will also be used later on when we solve equations

14. Simplify using chunking. Ex1. 4(x – 9) + 5(x – 9) – 13(x – 9) Ex2. 6(3x + 5) – (3x + 5) + 2(3x + 5) You can also use chunking to find values of expressions by determining the relationship between the original expression and the one in the question Ex3. If 2x = 23, find 6x Ex4. If 3y = 8, find 12y + 2 Sections from the book to read: 5-9

15. Section 6: Simplifying Rational Expressions Rational expressions contain fractions Remember that in order to add or subtract any fractions, they must have a common denominator When you find the common denominator multiply both the numerator and denominator by the same number Once the denominators are the same, add or subtract the numerators (combine like terms) and leave the denominator the same

16. Simplify each rational expression Ex1. Ex2. Ex3. Ex4. Sections from book to read: 3-9, 4-5, 5-9

17. Section 7: Multiplying with Monomials When you are multiplying terms, add the exponents of the variables that are alike Product of Powers Property: For all m and n, and all nonzero b, Simplify Ex1. Ex2. Ex3.

18. If you are raising a power to a power then you multiply exponents Power of a Power Property: For all m and n, and all nonzero b, Ex4. Simplify You distribute the exponent on the exterior of the parentheses to every part of the monomial within You CANNOT do this if there is any type of expression other than a monomial within the parentheses

19. Power of a Product Property: For all nonzero a and b, and for all n, Simplify Ex5. Ex6. Ex7. Solve for n. Sections from book to read: 2-5, 8-5, 8-8, 8-9

20. Section 8: Negative Exponents A negative exponent does NOT make anything in the expression negative Negative Exponent Property: For any nonzero b and all n, the reciprocal of Only the power with the negative exponent is changed Write with no negative exponents Ex1. Ex2. Ex3.

21. When you have a number raised to a negative exponent, use the negative to move the power to the opposite half of the fraction, then raise the base to the exponent Ex4. Write as a simple fraction Ex5. Write as a negative power of an integer The negative exponent property is one way to prove that any number to the zero power is equal to one (see page 516) Zero Exponent Property: If g is any nonzero real number, then

22. Ex6. Write without negative exponents Ex7. Simplify Ex8. Simplify Sections from book to read: 8-2, 8-6, 8-9, 12-7

23. Section 9: Division of Monomials When you divide monomials with matching variables, subtract the exponents Quotient of Powers Property: For all m and n, and all nonzero b, Read the directions to determine whether or not you can leave negative exponents in the answer, if you are unsure, write without negative exponents

24. Write as a simple fraction Ex1. Ex2. Simplify. Write the result as a fraction without any negative exponents Ex3.Ex4. Just like with a monomial being raised to a power, if you have a fraction being raised to a power you can distribute the exterior power

25. Power of a Quotient Property: For all nonzero a and b, and for all n, Write as a simple fraction Ex5.Ex6. Ex7. Sections from book to read: 8-7, 8-8, 8-9

26. Section 10: Multiplying and Dividing Rational Expressions Remember that when you multiply fractions you multiply numerators together and denominators together You can choose to reduce first or reduce after you multiply Multiplying Fractions Property: For all real numbers a, b, c, and d, with b and d nonzero,

27. When dividing fractions, flip the second fraction and then multiply Do not use mixed numbers with variables (i.e. or not Simplify. Write with no negative exponents. Ex1.Ex2. Ex3.Ex4. Sections from book to read: 2-3, 2-5

28. Section 11: Multiplying a Monomial by a Polynomial When you multiply a monomial by any other type of polynomial, you are distributing that monomial to each monomial in the polynomial Remember that you add exponents when you are multiplying Write your answers in standard form However many terms are in the polynomial is the number of terms in the answer

29. A subscript is a way of naming something, it is not a mathematical process i.e. x 1 is a way of naming the first x, like x 2 is a way of naming the second x Subscripts are written smaller and lower than the other numbers or variables Multiply. Ex1. 8x(5x³ + 4x² + 3x + 5) Ex2. Ex3. Sections from book to read: 3-7, 10-1, 10-3

30. Section 12: Multiplying a Binomial by a Binomial There is a mnemonic device that aids in remembering how to multiply a binomial by a binomial (it doesn’t work for anything else) F.O.I.L. stands for First, Outer, Inner, Last and it is the order that is commonly used when multiplying two binomials The FOIL algorithm is just an ordered way to use Extended Distributive Property

31. After multiplying using the FOIL algorithm, simplify if possible Multiply Ex1. (x + 3)(x + 7) Ex2. (x – 5)(x – 9) Ex3. (x – 7)(x + 8) Ex4. (2x + 3)(x – 6) Ex5. (3x – 8)(5x + 2) Ex6. (2x – 3y)(4x + 5y) Ex7. (6x² + 7)(4x² – 9) Section from the book to read: 10-5

32. Section 13: Special Binomial Products There are certain types of binomial products that have shortcuts you can use to multiply Square of a Sum is when you square a sum (addition problem) Square of a Difference is when you square a difference (subtraction problem) When you square a sum or a difference, the result is a perfect square trinomial Perfect Square Patterns: For all numbers a and b, (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b²

33. Expand Ex1. (x – 4)²Ex2. (m + 8)² Ex3. (2x + 1)²Ex4. (3a – 5)² You can use perfect square patterns to prove the Pythagorean Theorem is true (see page 648) If you have two binomials being multiplied that are nearly identical except one is a sum and one is a difference, the result is the difference of squares (because the inner and outer terms will cancel out) Difference of Two Squares Pattern: For all numbers a and b, (a + b)(a – b) = a² – b²

34. Expand Ex5. (x + 5)(x – 5)Ex6. (3x – 2)(3x + 2) You can use these two patterns to do some mental arithmetic Ex7. 53²Ex8. 81 · 79 Section from the book to read: 10-6

35. Section 14: Multiplying Polynomials When you are multiplying two polynomials together, you use the Extended Distributive Property to multiply every term in the 1 st polynomial by every term in the 2 nd polynomial and then simplify (if possible) See how to use rectangular visual displays to simplify this process on page 633 Develop an algorithm so that you do not miss any terms

36. Multiply Ex1. (x – 3)(4x³ + 3x² + 5x + 2) Ex2. (2y² + 3y + 4)(5y² + 6y – 3) Ex3. (3x + 5y + 7)(4x – 6y – 8) By the Commutative Property of Multiplication xy is the same as yx, but you should write the variables in alphabetical order Sections from the book to read: 2-1, 10-4

37. Section 15: Writing Expressions and Equations Key terms that mean to add: sum, plus, total, more than, in addition to, etc. Key terms that mean to subtract: difference, minus, less than, take away from, etc. Key terms that mean to multiply: times, product, of, multiplied by, etc. Key terms that mean to divide: divided by, quotient, etc. The word “is” means to put an equal sign there

38. Once something is referred to as “the quantity,” that should be placed in parentheses Write an expression for each sentence Ex1. The sum of 8 and the product of a number and 6 Ex2. The quantity of a number plus 7 will then be divided by 9 Ex3. The difference of 8 and a number You should also be able to write an expression or equation from a table

39. When you are studying a table, look for patterns (what can you do to the number on the left (or top) to get the number on the right (or bottom) that works every time) Ex4. Write an equation based on the information from the table Ex5. Pencils sell for $.24 each while notebooks sell for $.72 each. Write an expression to describe how to find the total cost if you buy p pencils and n notebooks. x1346810 y51114202632

40. Ex6. Steve charges a $40 consultation fee and then $10.50 per hour. Write an equation for Steve’s billing procedures. Ex7. A parking lot charges $3 for the first hour and then $2 for every hour after that. A)If a car is in the lot for 6 hours, how much will the owner pay? B)If a car is in the lot for h hours, how much will the owner pay? Write an equation. Sections of the book to read: 1-7, 1-9, 3-8