Unit 2Expressions

Like terms are those that have EXACTLY matching variables (order does not matter)

You can add and subtract the coefficients to like terms by using the distributive property in reverse

The Distributive Property: a(b + c) = ab + ac and (b + c)a = ab + ac

For example: 3x + 5x = (3 + 5)x = 8x Ex1. Simplify: 3a + 2b – 8a + b Ex2. Simplify

Section 1: Adding and Subtracting Like Terms

2 28 5 9 6x x x x

Ex3. Simplify

If it helps, you can change subtraction signs to adding negative values

Ex4. Simplify 10x – 8y – 4x – (-2y) If there is a negative or subtraction sign

directly outside a set of parentheses containing either a sum or difference, distribute the sign to each term within the parentheses

Ex5. Simplify 10x – (5x + 8) + 12 – 3x Ex6. Simplify (5n – 8p) – (9n – 5p) + 4p

13

2x x x

Opposite of a Sum Property: For all real numbers a and b, -(a + b) = -a + -b = -a – b

Opposite of Opposites Property (Op-op property): For a real number a, -(-a) = a

Opposite of a Difference Property: For all real numbers a and b, -(a – b) = -a + b

Ex7. Simplify

Ex8. Simplify

Sections from the book to read: 3-6 and 4-5

9 3 4 3 17 3

9 4 12m m m

A rational expression contains at least one fraction

You must have a common denominator in order to add or subtract fractions

Multiply the numerator and denominator of the fraction by the same number◦ Do this to both fractions so that the denominators

are the same Then add or subtract the numerators

(combining like terms) and leave the denominator the same

Section 2: Simplifying Rational Expressions

Simplify each rational expression Ex1.

Ex2.

Ex3.

Ex4.

Sections of the book to read: 3-9, 4-5, and 5-9

5 2 1

3 6

x x

3 1 5 3

8 6

m m

9 4 2

6 6

x x

x x

7 3 5

5 5

x x

x x

When you are multiplying terms, add the exponents of the variables that are alike

Product of Powers Property: For all m and n, and all nonzero b,

Simplify Ex1.

Ex2.

Ex3.

Section 3: Multiplying Monomials and Raising to a Power

m n m nb b b

5 4 2 3 83x y z x y z

2 3 5 64 6a b a b

2 5 3 26 3 7 2x x x x

When you raise a power to a power, multiply the exponents

Power of a Power Property: For all m and n, and all nonzero b,

Ex4. Simplify

If the exponent is directly outside of parentheses that contain a monomial, then you multiply every exponent inside of parentheses by the one outside

Power of a Product Property: For all nonzero a and b, and for all n,

nm mnb b

36x

n n nab a b

Simplify Ex5.

Ex6.

Ex7. Solve for n. Sections from the book to read: 2-5, 8-5, 8-8

and 8-9

23 4x y z

33 2 55a b cd

7 132 2 2n

A negative exponent does NOT make anything in the expression negative

Negative Exponent Property: For any nonzero b and all n, the reciprocal of

Only the power with the negative exponent is changed, it is moved to the other half of the fraction

Write with no negative exponents Ex1. Ex2. Ex3.

Ex4. Write as a simple fraction

Section 4: Negative Exponents

1nn

bb

nb

5x 3 4 2 55a b c d

2 3

5 6

a b

c d

43

Ex5. Write as a negative power of an integer

Zero Exponent Property: If g is any nonzero real number then,

Ex6. Write without negative exponents Ex7. Simplify

Ex8. Simplify

Sections from the book to read: 8-2, 8-6, 8-9, and 12-7

0 1g

234x

11

4

22

3

1

27

When dividing monomials, subtract the exponents of the matching variables

Quotient of Powers Property: For all m and n, and all nonzero b,

Write answers without negative exponents unless the directions allow it

Ex1. Simplify

Write as a simple fraction◦ Ex2. Ex3.

Section 5: Dividing Monomials and Raising to a Power

mm n

n

bb

b

8

3

x

x

13

9

5

5

12

15

4

4

Simplify. Write as a fraction with no negative exponents◦ Ex4. Ex5.

Power of a Quotient Property: For all nonzero a and b, and for all n,

Write as a simple fraction with no negative exponents◦ Ex6. Ex7. Ex8.

Sections from the book to read: 8-7, 8-8, 8-9

5 3 6

2 7 8

4

18

a b c

a b c

5 3 9

7 4

10

20

x y z

xy z

33

7

322

5

x

43

2

35

a

b

n n

n

a a

b b

To multiply rational expressions, multiply the numerators together and the denominators together and be sure to simplify◦ You can simplify before you multiply or after

To divide rational expressions, flip the second expression and then multiply

Do NOT use mixed numbers with variables◦ Yes: or No:

Section 6: Multiplying and Dividing Rational Expressions

29

4a b

29

4

a b 2124

a b

Simplify. Write the answer with no negative exponents.

Ex1. Ex2.

Ex3. Ex4.

Sections of the book to read: 2-3 and 2-5

2 4 7

4 3

2 6

3 5

a b a c

c b

3 5 3 6

4 9

5 4

8 15

x y y z

z x

3 5

5 6

2 8

7 3

d e d f

f e

5 3 7

4 3 6

4 6

5 7

a b a

c b c

Multiplying a monomial by a polynomial is using the distributive property

Write your answers in standard form A subscript is NOT a mathematical process, it is

just another name for a variable◦ i.e. x1 and x2 are two different variables

Multiply◦ Ex1.

◦ Ex2.

Section 7: Multiplying by Monomials and Binomials

5 34 3 8 7x x x x

2 3 2 4 53 5 6 3a b a b a b ab

If you are multiplying a binomial by another binomial, FOIL will help make sure you don’t miss any terms

FOIL: First, Inner, Outer, Last◦Multiply the First term in each binomial, then

multiply the two Inner terms, then multiply the two Outer terms, then multiply the two Last terms, and finally combine like terms

Multiply◦Ex3. (x + 4)(x + 6) Ex4. (m – 3)(m – 5)◦Ex5. (n + 6)(n – 9) Ex6. (2a + 3)(a

– 5)◦Ex7. (3w² + 5)(2w² ─ 7)

Sections of the book to read: 3-7, 10-1, 10-3, and 10-5

Use the Extended Distributive Property in order to multiply polynomials◦ Multiply every term in the first polynomial by every

term in the second polynomial See page 633 for a rectangular way to

demonstrate this property You can write the work vertically or horizontally

(your choice) Multiply

◦ Ex1.

◦ Ex2.

Section 8: Multiplying Polynomials

3 2( 5)(2 4 7 8)x x x x

2 2(2 3 7)(4 6)y y y y Sections of the book to read: 2-1 and 10-4

Perfect Square Patterns: For all numbers a and b (a + b)² = a² + 2ab + b² and (a – b)² = a² - 2ab + b²◦ You can use this shortcut when multiplying

Square of a sum is a sum squared◦ i.e. (a + b)²

Square of a difference is a difference squared◦ i.e. (a – b)²

The result of a square of a sum and a square of a difference is called a perfect square trinomial

Expand◦ Ex1. (x – 5)² Ex2. (a + 7)² Ex3. (4m – 3)²

Section 9: Special Binomial Products

If you multiply two binomials that are identical except one is addition and one is subtraction, the outer and inner terms will cancel out◦ The result is called the difference of squares

Difference of Two Squares Pattern: For all numbers a and b, (a + b)(a – b) = a² - b²

Expand◦ Ex4. (x + 5)(x – 5) Ex5. (3x – 2)(3x + 2)

You can use these patterns to do some basic arithmetic

Ex6. 43² Ex7. 81 · 79 Section of the book to read: 10-6

Once you read the word “is,” that is where you put the equal sign

If the book uses the word “the quantity,” that is where you put the parentheses

Write an expression for each sentence◦ Ex1. The sum of 8 and the product of a number and 6◦ Ex2. The quantity of a number plus seven will then be

divided by 9◦ Ex3. The difference of a 7 and a number

When given a table, look for a pattern to describe the situation

Section 10: Writing Expressions and Equations

Ex4. Write an equation based on the information

Ex5. Pencils sell for $0.24 each while notebooks sell for $0.72 each. Write an expression to describe how to find the total cost if you buy p pencils and n notebooks

Ex6. A parking lot charges $3 for the first hour and then $2 for every hour after that◦ A) If a car is in the lot for 6 hours, how much will the

owner pay?◦ B) If a car is in the lot for h hours, how much will the

owner pay? Sections of the book to read: 1-7, 1-9, and 3-8

x 1 3 4 6 8 10

y 5 11 14 20 26 32