Transcript Columbus State Community College Chapter 2 Section 2 Simplifying Expressions Ch 2 Sec 2: Slide #1 Simplifying Expressions 1.

Columbus State Community College
Chapter 2 Section 2
Simplifying Expressions
Ch 2 Sec 2: Slide #1
Simplifying Expressions
1. Combine like terms, using the distributive property.
2. Simplify expressions.
3. Use the distributive property to multiply.
Ch 2 Sec 2: Slide #2
Simplifying Expressions
The basic idea in simplifying expressions is to combine like terms
through addition and subtraction. Each addend and subtrahend in an
expression is a term.
Separates the terms
1 x + 4y – 5
Three terms
x is called a variable term. The coefficient is 1 and the variable is x.
4y is called a variable term. The coefficient is 4 and the variable is y.
5 ( or –5 ) is called a constant term.
Ch 2 Sec 2: Slide #3
Like Terms
Like Terms
Like terms are terms with exactly the same variable parts (the same
letters and exponents). The coefficients do not have to match.
Ch 2 Sec 2: Slide #4
Examples of Like Terms
Like Terms
1.
2x and –4x
Variable parts match; both are x.
2.
8b7 and 2b7
Variable parts match; both are b7.
3.
4.
–6a3c5
and 2a3c5
4 and –5
Variable parts match; both are a3c5.
No variable parts; numbers are like terms.
Ch 2 Sec 2: Slide #5
Examples of Unlike Terms
Unlike Terms
1.
4x and
–2x2
Variable parts do not match;
exponents are different.
2.
3a and 3b
Variable parts do not match;
letters are different.
3.
m2n3 and m2n5
Variable parts do not match;
exponents are different.
4.
7v and –5
Variable parts do not match; one term has a
variable part, but the other term does not.
Ch 2 Sec 2: Slide #6
Identifying Like Terms and Their Coefficients
EXAMPLE 1
Identifying Like Terms and Their Coefficients
List the like terms in each expression. Then identify the coefficients of
the like terms.
(a)
x + –7x2 + –4xy + –5x – 8
The like terms are x and –5x.
The coefficient of x is understood to be 1, and
the coefficient of –5x is –5.
Ch 2 Sec 2: Slide #7
Identifying Like Terms and Their Coefficients
EXAMPLE 1
Identifying Like Terms and Their Coefficients
List the like terms in each expression. Then identify the coefficients of
the like terms.
(b)
3a2b + 2ab2 – 4a2b + 8a2b2 – a2b
The like terms are 3a2b, 4a2b, and a2b.
The coefficient of 3a2b is 3,
the coefficient of 4a2b is 4 (or –4), and
the coefficient of a2b is 1 (or –1).
Ch 2 Sec 2: Slide #8
Identifying Like Terms and Their Coefficients
EXAMPLE 1
Identifying Like Terms and Their Coefficients
List the like terms in each expression. Then identify the coefficients of
the like terms.
(c)
8m + 2 + 9n + 6mn + 7mn2 – 5
The like terms are 2 and 5 (or –5).
The like terms are constants (there are no variable parts).
Ch 2 Sec 2: Slide #9
CAUTION
CAUTION
Notice that 4x and 7x is simplified to 11x, not to 11x2.
Variable part is unchanged.
Do not change x to x2.
Ch 2 Sec 2: Slide #10
Combining Like Terms
Combining Like Terms
Step 1 If there are any variable terms with no coefficient, write in
the understood 1.
Step 2 If there are any subtractions, change each one to adding
the opposite. Alternatively, you can treat each term that
follows a subtraction operator as a negative term.
Step 3 Find like terms (the variable parts match).
Step 4 Add (or subtract) the coefficients (number parts) of the
like terms. The variable part stays the same.
Ch 2 Sec 2: Slide #11
Combining Like Terms
EXAMPLE 2
Combining Like Terms
Combine like terms.
8a + 5a – a = 12a
(a)
8a + 5a –
a
8a + 5a –
1a
8a + 5a + –1a
(8
+ 5
No coefficient; write understood 1.
Change subtraction to addition of opposite.
Find like terms. Use the distributive property.
+ –1 ) a Add the coefficients.
12 a
The variable part, a, stays the same.
Ch 2 Sec 2: Slide #12
Combining Like Terms
EXAMPLE 2
Combining Like Terms
Combine like terms.
ALTERNATIVE METHOD
8a + 5a – a = 12a
(a)
8a + 5a –
a
8a + 5a – 1a
(8
+ 5
– 1)a
12 a
No coefficient; write understood 1.
Find like terms. Use the distributive property.
Add and subtract the coefficients.
The variable part, a, stays the same.
Ch 2 Sec 2: Slide #13
Combining Like Terms
EXAMPLE 2
Combining Like Terms
Combine like terms.
–3x2
(b)
– 4x2 = –7x2
–3x2
– 4x2
Change subtraction to addition of opposite.
–3x2
+ –4x2
Find like terms. Use the distributive property.
+ –4 ) x2
Add and subtract the coefficients.
–7 x2
The variable part, x2, stays the same.
( –3
Ch 2 Sec 2: Slide #14
Combining Like Terms
EXAMPLE 2
Combining Like Terms
Combine like terms.
–3x2
(b)
–3x2
( –3
ALTERNATIVE METHOD
– 4x2 = –7x2
– 4x2
Find like terms. Use the distributive property.
– 4 ) x2
Subtract the coefficients.
–7 x2
The variable part, x2, stays the same.
Ch 2 Sec 2: Slide #15
Simplifying Expressions
EXAMPLE 3
Simplifying Expressions
Simplify each expression by combining like terms.
(a)
2ac + 8c + 7ac = 9ac + 8c
2ac + 8c + 7ac
Rewrite expression using commutative property.
2ac + 7ac + 8c
Find like terms. Use the distributive property.
( 2 + 7 ) ac + 8c
9ac + 8c
Add inside parentheses.
The expression is simplified.
Ch 2 Sec 2: Slide #16
Simplifying Expressions
EXAMPLE 3
Simplifying Expressions
Simplify each expression by combining like terms.
(b)
3n – 4 – n + 9 = 2n + 5
3n – 4 –
n + 9
3n – 4 –
1n + 9
Change subtraction to adding the opposite.
3n + –4 + –1n + 9
Rewrite with like terms next to each other.
3n + –1n + –4 + 9
Use the distributive property.
( 3 + –1 )n + –4 + 9
2n + 5
Write 1 as the coefficient of n.
Combine like terms.
The expression is simplified.
Ch 2 Sec 2: Slide #17
Simplifying Expressions
EXAMPLE 3
Simplifying Expressions
Simplify each expression by combining like terms.
(b)
3n – 4 – n + 9 = 2n + 5
ALTERNATIVE METHOD
3n – 4 –
n + 9
Write 1 as the coefficient of n.
3n – 4 – 1n + 9
Rewrite with like terms next to each other.
3n – 1n – 4 + 9
Combine like terms.
2n + 5
The expression is simplified.
Ch 2 Sec 2: Slide #18
Note on the Order of Listing Terms
NOTE
When combining like terms, we typically write the variable terms in
alphabetical order. A constant term (number only) will be written last.
So, in Examples 3(a) and 3(b), the preferred and alternative ways of
writing the expressions are as follows:
The simplified expression is 9ac + 8c (alphabetical order).
However, by the commutative property of addition, 8c + 9ac is also
correct.
The simplified expression is 2n + 5 (constant written last).
However, by the commutative property of addition, 5 + 2n is also
correct.
Ch 2 Sec 2: Slide #19
Simplifying Multiplication Expressions
EXAMPLE 4
Simplifying Multiplication Expressions
Simplify.
(a)
4 ( –9x ) = –36x
4 • ( –9 • x )
Rewrite expression using associative property.
4 • ( –9 • x )
Multiply.
–36
x
The expression is simplified.
Ch 2 Sec 2: Slide #20
Simplifying Multiplication Expressions
EXAMPLE 4
Simplifying Multiplication Expressions
Simplify.
–3
(b)
( –2n ) = 6n
–3
• ( –2 • n )
Rewrite expression using associative property.
–3
• ( –2 • n )
Multiply.
6n
The expression is simplified.
Ch 2 Sec 2: Slide #21
Simplifying Multiplication Expressions
EXAMPLE 4
Simplifying Multiplication Expressions
Simplify.
–5
(c)
( 8y2 ) = –40y2
–5
• ( 8 • y2 )
Rewrite expression using associative property.
–5
• ( 8 • y2 )
Multiply.
–40 y2
The expression is simplified.
Ch 2 Sec 2: Slide #22
The Distributive Property
Multiplication distributes over addition and subtraction as follows:
2 ( x + 7 ) can be written as 2 • x + 2 • 7
So, 2 ( x + 7Stays
) simplifies
to 2x 2x
+ 14.+ 14
as addition
6 ( x – 3 ) can be written as 6 • x – 6 • 3
as subtraction
So, 6 ( x – Stays
3 ) simplifies
to 6x 6x
– 18.–
18
Ch 2 Sec 2: Slide #23
Using the Distributive Property
EXAMPLE 5
Using the Distributive Property
Simplify.
(a)
5 ( 2a – 3 ) can be written as 5 • 2a – 5 • 3
Stays as subtraction10a
–
15
So, 5 ( 2a – 3 ) simplifies to 10a – 15.
Ch 2 Sec 2: Slide #24
Using the Distributive Property
EXAMPLE 5
Using the Distributive Property
Simplify.
(b)
8 ( 4n + 7 ) can be written as 8 • 4n + 8 • 7
Stays as addition 32n
+ 56
So, 8 ( 4n + 7 ) simplifies to 32n + 56.
Ch 2 Sec 2: Slide #25
Using the Distributive Property
EXAMPLE 5
Using the Distributive Property
Simplify.
(c)
–3
( 9k + 2 ) can be written as –3 • 9k + –3 • 2
Stays as addition
–27k
+
–6
So, –3 ( 9k + 2 ) simplifies to –27k + –6.
Using the definition of subtraction “in reverse”, we rewrite
–27k + –6
–27k – 6.
as
Ch 2 Sec 2: Slide #26
Using the Distributive Property
EXAMPLE 5
Simplify.
(c)
–3
Using the Distributive Property
ALTERNATIVE METHOD
( 9k + 2 ) = –27k – 6
–3(distribute,
A
When
negative
you
•–9k
3 ) x a “positive”
treat the operation
( + 2 ) = within
“negative” 6 ( – 6 ).
parentheses as the sign of the second term. In this
–3 ( 9k + 2 ) = –27k– – 6
example, as we distribute the 3 to the 2, we read
it as “ –3 times positive 2”.
–3 • + 2
Ch 2 Sec 2: Slide #27
Simplifying a More Complex Expression
EXAMPLE 6
Simplifying a More Complex Expression
7 + 4 ( x – 5 ) = 4x – 13
Simplify:
7 + 4(x – 5)
Do not add 7 + 4. Use the distributive property.
7 + 4•x – 4•5
Do the multiplication.
7 + 4x – 20
Rewrite so that like terms are next to each other.
4x +
7 – 20
4x +
4x – 13
–13
Subtract 7 – 20.
Rewrite using the definition of subtraction “in
reverse”.
Ch 2 Sec 2: Slide #28
Simplifying Expressions
Chapter 2 Section 2 – Completed
Written by John T. Wallace
Ch 2 Sec 2: Slide #29