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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.
27,7
5
3
1,4,75 2 xxyxx
Consider the example:
The terms of the expression are separated by addition. There are 3 terms in this example and they are
.
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.
75 2 xx
7,,5 2 xx
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
a ( b + c ) = ba + ca
To simplify some expressions we may need to use the Distributive Property
Do you remember it?
Distributive Property
Examples
Example 1: 6(x + 2)
Distribute the 6.
6 (x + 2) = x(6) + 2(6)
= 6x + 12
Example 2: -4(x – 3)
Distribute the –4.
-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)
= 1( x – 2 )
= x(1) – 2(1)
= x - 2
Notice multiplying by a 1 does nothing to the expression in the ( )’s.
Example 4: -(4x – 3)
= -1(4x – 3)
= 4x(-1) – 3(-1)
= -4x + 3
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
-6.7xy , 02xy
The variable factors are
identical.
Unlike Terms
5x , 8y
The variable factors are
not identical.
22 8,3 xyyx
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
31316
terms.likeCombine
31334124
terms.theReorder
33124134
2
22
22
yxx
yxxxx
xxxyx
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
431062
1 xx
Simplifying Example
Distribute. 43106
2
1 xx
Simplifying Example
Distribute. 43106
2
1 xx
12353
3432
110
2
16
xx
xx
Simplifying Example
Distribute.
Combine like terms.
431062
1 xx
12353
3432
110
2
16
xx
xx
Simplifying Example
Distribute.
Combine like terms.
431062
1 xx
12353
3432
110
2
16
xx
xx
76 x
Evaluating Expressions
Remember to use correct order of operations.
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.
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
1and2when34Evaluate 22 yxyxyx
Evaluating Example
Substitute in the numbers.
1and2when34Evaluate 22 yxyxyx
Evaluating Example
Substitute in the numbers.
1and2when34Evaluate 22 yxyxyx
22 131242
Evaluating Example
Remember correct order of operations.
1and2when34Evaluate 22 yxyxyx
22 131242
Substitute in the numbers.
131244
384
15
Common Mistakes
Incorrect Correct
Your Turn
• Find the product
1. (-8)(3)
2. (20)(-65)
3. (-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. -24
2. -1300
3. -9
4. 3y
5. -5a3
6. -48
7. 12
8. -28
9. -27
10. 76
Find the product.
a. (9)(–3) b.
c. (–3)3 d.
1(8) ( 6)
2
1( 2) ( 3)( 5)
2
-27(–4)(–6)
24
(–3)(–3)(–3)(9)(–3)
–27
1(–3)(–5)(–3)(–5)
15
Find the product.
a. (–n)(–n)
b. (–4)(–x)(–x)(x)
c. –(b)3
d. (–y)4
Two negative signs: n2
Three negative signs: –4x3
One negative sign: –(b)(b)(b) = –b3
Four negative signs: (–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 3Evaluate the expression when x = –7.a. 2(–x)(–x)
2 ( 7) ( 7)
2 7 7
14 7
98
OR simplify first:
2(–x)(–x)
2x2
2(-7)2
2(49)98
Extra Example 3 (cont.)Evaluate the expression when x = –7.
b. 25
7x
25( 7)
7
25 7
7
5(2)
10
25
7x
25( 7)
7 2
357
10
OR use the associative property:
CheckpointFind the product.1. (–2)(4.5)(–10) 2. (–4)(–x)2
3. Evaluate the expression when x = –3:(–1• x)(x)
90 –4x2
–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.
acabcba
5323523
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
unique real number such that their
product is 1.
1
a
11
a
a
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
3 + 7 = 7 + 3
Commutative Commutative Property of AdditionProperty of Addition
2.2.
8 + 0 = 8
Identity Property of Identity Property of AdditionAddition
3.3.
6 • 4 = 4 • 6
Commutative Property Commutative Property of Multiplicationof Multiplication
5.5.
5 • 1 = 5
Identity Property of Identity Property of MultiplicationMultiplication
11.11.
51/7 + 0 = 51/7
Identity Property of Identity Property of AdditionAddition
25.25.
a + (-a) = 0
Inverse Property of Inverse Property of AdditionAddition
40.40.