Download Unit 2: Expressions

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
• 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
4x  9
y 3
when x = 6 and y = 5
• Ex2. Evaluate 4  9  3  2  4 3  48
6
• Ex3. Evaluate  8  4x 
6
when x = -3
• Sections from the book to read: 1-1 and 1-4
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)
•
•
•
•
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)
5
6x yz
3
B)
4
5
15abc 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) 8 x  4 xy  5 x 2 y 3 B) 3a 2b 4
 6ab  2a b
3
3 3
• 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
• 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
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  8 x 2  5 x    9 x  6 x 2 
• Notice that the exponents do not change when
you add (or subtract) like terms
1
• Ex3. Simplify x  3x  x
2
• Ex4. Simplify -3x + 2y + 8x
• Section from the book to read: 3-6
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. 9 3  4 3  17 3
• Ex7. 9 m  4 m  12 m
• Sections from the book to read: 4-5
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. 5 x  2 x  1
3
6
• Ex2. 3m  1  5m  3
8
6
• Ex3.
9 x 3x  2

x 6 x  6
• Ex4.
7 x 3x  5

x 5 x 5
• Sections from book to read: 3-9, 4-5, 5-9
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, b m  b n  b m  n
• Simplify
5
4
2
3
8
• Ex1. 3x  y  z  x  y  z
• Ex2. 4a 2b3  6a 5b 6
• Ex3. 6 x 2  3x5  7 x3  2 x 2 
• If you are raising a power to a power then you
multiply exponents
• Power of a Power Property: For all m and n, and
n
m
all nonzero b,  b   bmn
6 3
• Ex4. Simplify  x 
• 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
• Power of a Product Property: For all nonzero a
n
n n
ab

a
b


and b, and for all n,
• Simplify
2
3 4
• Ex5.  x y z 
• Ex6.  5a b cd
3 2

5 3
• Ex7. Solve for n. 27  2n  213
• Sections from book to read: 2-5, 8-5, 8-8, 8-9
Section 8: Negative Exponents
• A negative exponent does NOT make anything in
the expression negative
• Negative Exponent Property: For any nonzero b
n
1

n
and all n, b 
the reciprocal of b
bn
• Only the power with the negative exponent is
changed
• Write with no negative exponents
2 3
5
3 4 2
a
• Ex1. x
Ex2. 5a b c d
Ex3. b
c 5 d 6
• 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
4
• Ex4. Write as a simple fraction 3
1
• Ex5. Write as a negative power of an integer 27
• 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 g 0  1
• Ex6. Write without negative exponents  4x 3 2
1
• Ex7. Simplify
• Ex8. Simplify
1
 
4
2
 
3
2
• Sections from book to read: 8-2, 8-6, 8-9, 12-7
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, b m
mn
b
n
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
• Write as a simple fraction
13
12
5
4
• Ex1.
Ex2.
15
59
4
• Simplify. Write the result as a fraction without
any negative exponents
3
9
4 5
• Ex3. 4a 2b9 c3
Ex4. 10 x7 yz
4 10
18a b c
20 x y z
• 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
• Power of a Quotient Property: For all nonzero a
n
n
and b, and for all n,  a  a
   n
b
b
• Write as a simple fraction
3
2 3
• Ex5.  3 
Ex6.  2 x 
 5 
7
3a 

• Ex7. 5   
 b 
4
• Sections from book to read: 8-7, 8-8, 8-9
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.
9 2
1a 2b
9a b
a b or
not
2
4
4
4
• Simplify. Write with no negative exponents.
4 2
2 8
2 3
2
3
x
y
4
x
z
• Ex1. 6a b  3ab
Ex2.

5
7
3
8
z
15
y
11c 10c
2
• Ex3. 2 xyz 6a 2 y 3
Ex4. 12d 4 e 2 9a 5 d


3
6
3 4
15a e
2e
9abc 24b c
• Sections from book to read: 2-3, 2-5
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)
2
4
2
ab
3
a
b

6
ab


• Ex2.
• Ex3. 4 x 2 y 3  2 x 4 y 2  3x 2 y  5 xy 
• Sections from book to read: 3-7, 10-1, 10-3
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
x
1
3
4
6
8
10
y
5
11
14
20
26
32
• 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.
• 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