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Precalculus
Chapter P.1 Part 2 of 3
Mr. Chapman
Manchester High School

Algebraic Expressions

Evaluating Algebraic Expressions

Using the Basic Rules and Properties of
Algebra
Definition of an Algebraic Expression:
A collection of letters (variables) and real
numbers (constants) combined using:
Addition
Subtraction
Multiplication
Division
Exponentiation
Examples:
1 term
5x
2x  3
7x  y
1 term
2 terms
2 terms
4
x2  2
Terms of an Algebraic Expression:
Individual terms are separated by addition
or subtraction signs.
How many terms in each of the algebraic
expressions above?
Consider the polynomial:
x  5x  8
2
How many terms? 3
variable terms: 2
constant terms: 1
Coefficients are the numerical factor (multi2
plier) in a variable term. 1? x  (5) x  8
Challenge Question!
Given the polynomial x  5x  8
What is the number 8?
A: a coefficient
C: both A & B
B: a constant term
D: not A or B
2
Does it multiply a variable? No
Is it a term by itself? YES!!

Algebraic Expressions

Evaluating Algebraic Expressions

Using the Basic Rules and Properties of
Algebra
To evaluate Algebraic Expressions:
Replace the variables with their assumed
or desired numerical value.
This is called the Substitution Principle.
Examples:
Value of
Expression Variable
 3x  5
3x  2 x  1
2
x3
x  1
Substitution
 3(3)  5
95 
Expr.
Value
4
3(1)  2(1)  1
2
3  2 1 
0

Algebraic Expressions

Evaluating Algebraic Expressions

Using the Basic Rules and Properties of
Algebra

Four Arithmetic Operations:
 Addition (+)
 Subtraction (-)
 Multiplication ( or  )
Multiplicative
12
1
Inverse
 12 
 Division (  or  )
a  b  a  (b)
5  3  5  (3)
Additive Inverse
a
1
 a 
b
b
Primary
Operations
Addition
Multiplication
4
4
Secondary
(Inverse)
Operations
Subtraction
Division
Let a, b, and c be real numbers, variables,
or expressions.
5,
x,
3x – 2
Rules and Properties of Algebra are only
given for the Primary Operations.
They are considered to also apply to
secondary (inverse) operations by
extension.
Commutative Property of Addition
ab  ba
4x  x2  x2  4x
Memory Tip:
Commute means travel (as in “commute to
work”). Note that the terms in this property
exchange places or “travel”.
Commutative Property of Multiplication
a b  b a
( 4  x) x  x ( 4  x)
2
2
Memory Tip:
Commute means travel (as in “commute to
work”). Note that the terms in this property
exchange places or “travel”.
Associative Property of Addition
(a  b)  c  a  (b  c)
( g  b) ( x( y5)w) x gx(b (5 yx w))
2
2
Memory Tip:
Terms do not commute or “travel”.
(That is, their order stays the same!)
Terms change who they associate with (are
grouped with) by using parentheses.
Associative Property of Multiplication
(a  b)c  a(b  c)
(2 x  3 y)8  2 x(3 y  8)
Memory Tip:
Terms do not commute or “travel”.
(That is, their order stays the same!)
Terms change who they associate with (are
grouped with) by using parentheses.
Distributive Property
a(b  c)  a  b  a  c
3x(5  2 x)  3x  5  3x  2 x
Memory Tip:
Both addition and multiplication involved.
The term outside the parentheses gets
distributed to (or shared with) every term
inside the parentheses. (Like treats!)
Distributive Property
(a  b)c  a  c  b  c
( y  8) y  y  y  8  y
Memory Tip:
Both addition and multiplication involved.
The term outside the parentheses gets
distributed to (or shared with) every term
inside the parentheses. (Like treats!)
Distributive Property
(a  b)c  a  c  b  c
( y  8) y  y  y  8  y
( y  8) y  y  y  8  y
Memory Tip:
Both addition and multiplication involved.
The term outside the parentheses gets
distributed to (or shared with) every term
inside the parentheses. (Like treats!)
Additive Identity Property
a0  a
5y  0  5y
2
2
Memory Tip:
In math, an identity is true for all values.
Identity also means the state of being the
same. In this case the value of “a” is
unchanged.
Multiplicative Identity Property
a 1  a
4x 1  4x
2
2
Memory Tip:
In math, an identity is true for all values.
Identity also means the state of being the
same. In this case the value of “a” is
unchanged.
Additive Inverse Property
a  ( a )  0
5x  (5x )  0
3
3
Memory Tip:
The inverse of addition is subtraction.
Subtraction is adding the same value but
of the opposite sign.
Multiplicative Inverse Property
1
a     1, a  0
a
 1 
2
( x  4) 2
 1
 x 4
Memory Tip:
The inverse of multiplication is division.
Division is multiplying by one over
something.
Homework:
Pg. 10: 67-111 Odd