Unit 2: ExpressionsSection 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

• 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

• 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 9

3

x

y

3 484 9 3 2 4

6

68 4x

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)

• 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) 5 36x yz 4 515abc d e

• 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

2 38 4 5x xy x y 2 4 3 3 33 6 2a b ab a b

• Ex4. Name the type of expression (monomial, binomial, trinomial, other)a) 5x + 3y – 2 b) 9a + 12c) 8c + 9d + 2e – 5f

d) 3xyz• Sections from the book to read: 3-6 and 10-1

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

• 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

2 28 5 9 6x x x x

13

2x x x

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

• 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

• 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

9 3 4 3 17 3

9 4 12m m m

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

• 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

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

• Simplify each rational expression• Ex1.

• Ex2.

• Ex3.

• Ex4.

• Sections from book to read: 3-9, 4-5, 5-9

5 2 1

3 6

x x

3 1 5 3

8 6

m m

9 3 2

6 6

x x

x x

7 3 5

5 5

x x

x x

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.

m n m nb b b

5 4 2 3 83x y z x y z 2 3 5 64 6a b a b

2 5 3 26 3 7 2x x x x

• 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

nm mnb b 36x

• 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

23 4x y z

33 2 55a b cd

7 132 2 2n

n n nab a b

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.

1nn

bb

nb

5x 3 4 25a b c d 2 3

5 6

a b

c d

• 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

431

27

0 1g

• Ex6. Write without negative exponents

• Ex7. Simplify

• Ex8. Simplify

• Sections from book to read: 8-2, 8-6, 8-9, 12-7

234x

11

4

22

3

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

mm n

n

bb

b

• 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

13

9

5

5

12

15

4

4

4 5

2 9 3

4

18

a b c

a b c

3 9

7 4 10

10

20

x yz

x y z

• 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

n n

n

a a

b b

33

7

322

5

x

43

5a

b

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,

a c ac

b d bd

• 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

29

4a b

29

4

a b21

24

a b

2 3 2

3

6 3

11 10

a b ab

c c

4 2 2 8

5 7

3 4

8 15

x y x z

z y

2 3

3 4

2 6

9 24

xyz a y

abc b c

4 2 5

3 6

12 9

15 2

d e a d

a e e

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

• A subscript is a way of naming something, it is not a mathematical process

• i.e. x1 is a way of naming the first x, like x2 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

2 42 3 6ab a b ab

2 3 4 2 24 2 3 5x y x y x y xy

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

• 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

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

• Expand• Ex1. (x – 4)² Ex2. (m + 8)²• Ex3. (2x + 1)² Ex4. (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

• Expand• Ex5. (x + 5)(x – 5) Ex6. (3x – 2)(3x + 2)• You can use these two patterns to do some

mental arithmetic• Ex7. 53² Ex8. 81 · 79• Section from the book to read: 10-6

Section 14: Multiplying Polynomials• When you are multiplying two polynomials

together, you use the Extended Distributive Property to multiply every term in the 1st polynomial by every term in the 2nd 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

• 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

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

• 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

• 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.

x 1 3 4 6 8 10

y 5 11 14 20 26 32

• 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