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Transcript of Expressions. LLike terms are those that have EXACTLY matching variables (order does not matter) YYou...
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