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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