Download Distributive Property

Chapter 2 Review
Algebra 1
Algebraic Expressions
An algebraic expression is a collection of real
numbers, variables, grouping symbols and
operation symbols.
Here are some examples of algebraic expressions.
1
5
5x  x  7 , 4 ,
xy  ,
3
7
2
 7 x  2 
Consider the example: 5 x  x  7
2
The terms of the expression are separated by addition.
There are 3 terms in this example and they are
5x 2 , x ,  7 .
The coefficient of a variable term is the real number
factor. The first term has coefficient of 5. The second
term has an unwritten coefficient of 1.
The last term , -7, is called a constant since there is no
variable in the term.
Let’s begin with a review of two important
skills for simplifying expression, using the
Distributive Property and combining like
terms. Then we will use both skills in the
same simplifying problem.
Distributive Property
To simplify some expressions we may
need to use the Distributive Property
Do you remember it?
Distributive Property
a ( b + c ) = ba + ca
Examples
Example 1: 6(x + 2)
Distribute the 6.
Example 2: -4(x – 3)
Distribute the –4.
6 (x + 2) = x(6) + 2(6)
= 6x + 12
-4 (x – 3) = x(-4) –3(-4)
= -4x + 12
Practice Problem
Try the Distributive Property on -7 ( x – 2 ) .
Be sure to multiply each term by a –7.
-7 ( x – 2 ) = x(-7) – 2(-7)
= -7x + 14
Notice when a negative is distributed all the signs of the
terms in the ( )’s change.
Examples with 1 and –1.
Example 3: (x – 2)
Example 4: -(4x – 3)
= 1( x – 2 )
= -1(4x – 3)
= x(1) – 2(1)
= 4x(-1) – 3(-1)
=x - 2
= -4x + 3
Notice multiplying by a 1 does
nothing to the expression in the
( )’s.
Notice that multiplying by a –1
changes the signs of each term
in the ( )’s.
Like Terms
Like terms are terms with the same
variables raised to the same power.
Hint: The idea is that the variable part of
the terms must be identical for them to be
like terms.
Examples
Like Terms
5x , -14x
Unlike Terms
5x , 8y
-6.7xy , 02xy
 3x y , 8xy
The variable factors are
identical.
2
2
The variable factors are
not identical.
Combining Like Terms
Recall the Distributive Property
a (b + c) = b(a) +c(a)
To see how like terms are combined use the
Distributive Property in reverse.
5x + 7x = x (5 + 7)
= x (12)
= 12x
Example
All that work is not necessary every time.
Simply identify the like terms and add their
coefficients.
4x + 7y – x + 5y = 4x – x + 7y +5y
= 3x + 12y
Collecting Like Terms Example
4 x 2  13 y  4 x  12 x 2  3 x  3
Reorder the terms.
4 x 2  12 x 2  4 x  3 x  13 y  3
Combine like terms.
16 x  x  13 y  3
2
Both Skills
This example requires both the Distributive
Property and combining like terms.
5(x – 2) –3(2x – 7)
Distribute the 5 and the –3.
x(5) - 2(5) + 2x(-3) - 7(-3)
5x – 10 – 6x + 21
Combine like terms.
- x+11
Simplifying Example
1
6 x  10  3x  4
2
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
1
1
6 x   10   x3  43
2
2
3x  5  3x 12
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
1
1
6 x   10   x3  43
2
2
3x  5  3x 12
Combine like terms.
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
1
1
6 x   10   x3  43
2
2
3x  5  3x 12
Combine like terms.
6x  7
Evaluating Expressions
Evaluate the expression 2x – 3xy +4y when
x = 3 and y = -5.
To find the numerical value of the expression,
simply replace the variables in the expression
with the appropriate number.
Remember to use correct order of operations.
Example
Evaluate 2x–3xy +4y when x = 3 and y = -5.
Substitute in the numbers.
2(3) – 3(3)(-5) + 4(-5)
Use correct order of operations.
6 + 45 – 20
51 – 20
31
Evaluating Example
Evaluate x 2  4 xy  3 y 2 when x  2 and y  1
Evaluating Example
Evaluate x  4 xy  3 y when x  2 and y  1
2
2
Substitute in the numbers.
Evaluating Example
Evaluate x  4 xy  3 y when x  2 and y  1
2
2
Substitute in the numbers.
22  42 1  3 12
Evaluating Example
Evaluate x 2  4 xy  3 y 2 when x  2 and y  1
Substitute in the numbers.
22  42 1  3 12
Remember correct order of operations.
4  421  31
483
15
Common Mistakes
Incorrect
Correct
Your Turn
•
1.
2.
3.
Find the product
(-8)(3)
(20)(-65)
(-15)
• Simplify the variable expression
4. (-3)(-y)
5. 5(-a)(-a)(-a)
Your Turn
• Evaluate the expression:
6. -8x
when x = 6
7. 3x2
when x = -2
8. -4(|y – 12|)
when y = 5
9. -2x2 + 3x – 7
when x = 4
10. 9r3 – (- 2r)
when r = 2
Your Turn Solutions
1.
2.
3.
4.
5.
-24
-1300
-9
3y
-5a3
6. -48
7. 12
8. -28
9. -27
10. 76
Find the product.
a. (9)(–3)
-27
 1
b. (8)    ( 6)
 2
(–4)(–6)
24
c.
(–3)3
(–3)(–3)(–3)
(9)(–3)
–27
 1
d. ( 2)    ( 3)( 5)
 2
1(–3)(–5)
(–3)(–5)
15
Find the product.
a. (–n)(–n)
Two negative signs:
b. (–4)(–x)(–x)(x)
n2
Three negative signs:
c. –(b)3
One negative sign:
d. (–y)4
Four negative signs:
–4x3
–(b)(b)(b) = –b3
(–y)(–y)(–y)(–y) = y4
SUMMARY: An even number of negative signs results in a
positive product, and an odd number of negative
signs results in a negative product.
Extra Example 3
Evaluate the expression when x = –7.
a. 2(–x)(–x)
2  ( 7)  ( 7) 
OR simplify first:
2  7  7 
14  7 
98
2(–x)(–x)
2x2
2(-7)2
2(49)
98
Extra Example 3 (cont.)
Evaluate the expression when x = –7.
b.
 2

5

x

  
 7
 2
 5( 7)   
 7
 2
35   
 7
10
OR use the associative property:
 2

5

x

  
 7
 2

5(

7)

  
 7
  2 
5  7    
  7 
5(2)
10
Checkpoint
Find the product.
1. (–2)(4.5)(–10)
90
2. (–4)(–x)2
–4x2
3. Evaluate the expression when x = –3:
(–1• x)(x)
–9
Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×
Commutative Properties
• Changing the order of the
numbers in addition or
multiplication will not change the
result.
• Commutative Property of Addition
states: 2 + 3 = 3 + 2 or a + b = b +
a.
• Commutative Property of
Multiplication states: 4 • 5 = 5 • 4
or ab = ba.
Associative Properties
• Changing the grouping of the
numbers in addition or multiplication
will not change the result.
• Associative Property of Addition
states: 3 + (4 + 5)= (3 + 4)+ 5 or
a + (b + c)= (a + b)+ c
• Associative Property of Multiplication
states: (2 • 3) • 4 = 2 • (3 • 4) or
(ab)c = a(bc)
Distributive Property
Multiplication distributes over addition.
ab  c  ab  ac
32  5  3  2  3  5
Additive Identity Property
• There exists a unique number 0
such that zero preserves identities
under addition.
a + 0 = a and 0 + a =
a
• In other words adding zero to a
number does not change its value.
Multiplicative Identity Property
• There exists a unique number 1
such that the number 1 preserves
identities under multiplication.
a ∙ 1 = a and 1 ∙ a = a
• In other words multiplying a
number by 1 does not change the
value of the number.
Additive Inverse Property
• For each real number a there
exists a unique real number –a
such that their sum is zero.
a + (-a) = 0
• In other words opposites add to
zero.
Multiplicative Inverse
Property
• For each real number a there exists a
1
unique real number
a
product is 1.
1
a 1
a
such that their
Let’s play “Name that property!”
State the property or properties
that justify the following.
3+2=2+3
Commutative Property
State the property or properties
that justify the following.
10(1/10) = 1
Multiplicative Inverse Property
State the property or properties
that justify the following.
3(x – 10) = 3x –
30
Distributive Property
State the property or properties
that justify the following.
3 + (4 + 5) = (3 + 4) +
5
Associative Property
State the property or properties
that justify the following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative Property
2.
3+7=7+3
Commutative
Property of Addition
3.
8+0=8
Identity Property of
Addition
5.
6•4=4•6
Commutative Property
of Multiplication
11.
5•1=5
Identity Property of
Multiplication
25.
1
5/
+
0
=
7
1
5/
7
Identity Property of
Addition
40.
a + (-a) = 0
Inverse Property of
Addition