Chapter 2 Review Algebra 1 Algebraic Expressions An algebraic expression is a collection of real...

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.