FUNDAMENTAL CONCEPTS OF ALGEBRA Unit 1
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FUNDAMENTAL CONCEPTS OF ALGEBRA
Unit 1
EXPONENTS AND RADICALS
Lesson 1.2
Lesson Essential Question (LEQ)
How do we simplify radicals and algebraic expressions with rational exponents?
Laws of Exponents for Real Numbers
A)
B)
C)
D)
E)
**(Blue Tables on Page 20)**(Always simplify negative exponents completely!)
Fractional Exponents
or
**(Green Table on Page 27)**
Simplifying Expressions with Exponents
Examples:
Textbook Page 29 #’s 16, 22, 32, 38, 44, 46
Homework:
Page 29 #’s 11-45 odds only
Bell Work:
Simplify:
Radicals
The whole thing is called a radical.
The n represents the index or root of the radical.
The a represents the radicand.
You need to know:
Squares:
Cubes:
Prime Factorization:
Laws of Radicals
Page 24 in your textbook.
These are properties and laws of radicals that you should file into your long-term memory!!!
Removing/Simplifying nth Powers
How would we simplify the following?
A)
B)
C)
Rationalizing Denominators
It is impossible to divide by an irrational number (which most radicals are), so we have to rationalize the denominator of any fractional expression that has a radical in it. **(Blue Table on Page 26)**
Ex:
Ex: :
Homework:
Pages 29-30 #’s 53-79 odds only
Bell Work:
Simplify each expression completely:
1)
2)
Class Examples:
Textbook Pages 29-30 #’s 24, 30, 44, 64, 70, 74, 76, 78, 80, 100
Homework:
Review Blue Tables on Pages 32, 36, and 38.
Bell Work:
Simplify the expression:
2)
ALGEBRAIC EXPRESSIONS
Lesson 1.3
Lesson Essential Question (LEQ)
How do we perform operations and factor algebraic expressions?
Polynomials
Adding
Subtracting
Multiplying
Dividing
Polynomial Product Formulas
**(Blue Table on Page 36)**
Hint: If you know these, it will save you time!
Polynomial Examples
Textbook Page 43 #’s 4, 10, 14, 18, 28, 34
Homework:
Pages 43-44 #’s 1-21 odds, 25, 29, 33, 35, 37, 39
Bell Work:
Simplify:
Factoring Polynomials
To factor any polynomial, you must first take out the Greatest Common Factor (GCF).
Factor:
When you factor something, you are breaking it down to its original factors.
Examples Page 44 #’s 46 and 50
Trinomials
In the form , when a = 1. (simple)Ex:
In the form , when a≠ 1. (Master Product Method)
Ex:
Examples Page 44 #’s 54 and 58
Special Case Binomials:
Difference of Two Squares
Difference of Two Cubes
Sum of Two Cubes
**(Blue Table on Page 38)**
Examples Page 44 #’s 72, 76, 78, 80
Factoring By Grouping
When you have four different terms that are separated by addition or subtraction, you can group certain terms together in order to factor.
Examples Page 44 #’s 86, 88
Remember:
It is possible that a polynomial cannot be factored, so in this instance, you would say the polynomial expression is irreducible.
Ex: , this is irreducible.
Homework:
Page 44 #’s 47, 51, 55, 57, 61, 69, 75, 81, 85, 89, 95
What you need to know for the quiz tomorrow:Powers and RadicalsAdd/Sub/Mult/Div PolynomialsFactoring
Look through the examples we did in class and the problems from previous homework assignments and you should be fine!
Bonus for the Quiz:
Bell Work:
Factor:
FRACTIONAL EXPRESSIONS
Lesson 1.4
Lesson Essential Question (LEQ)
How do we perform operations with fractional expressions?
Things to remember:
A fractional expression is a quotient of two algebraic expressions.
Whenever we have a variable in the denominator, we are going to have excluded values. Why?
Examples of Multiplying/Dividing
Page 54 #’s 8, 12, 14
Adding and Subtracting Fractional Expressions
MUST HAVE A COMMON DENOMINATOR!!!
If it does not have a common denominator, find it!
Examples: Page 54 #’s 20, 22, 26
Homework:
Page 54 #’s 5 – 27 odds only
Bell Work:
Simplify:
Complex Fractions:
A complex fraction is when you have a quotient in which the numerator and/or the denominator is a fractional expression.
We need to simplify the numerator and denominator as much as possible before we divide them!
Examples: Page 54 #’s 34, 40, 44
Remember that division is the same as multiplying by the reciprocal!!!!!!!!!!!
Rationalizing
Sometimes it is necessary to rationalize a denominator or numerator of a fractional expression.
We multiply the numerator and denominator by the conjugate.
Examples: Page 55 #’s 52, 56, 60
Homework:
Pages 54-55 #’s 33, 39, 43, 49, 51, 55
Bell Work:
Simplify:
1)
2)
Class Work:
Pages 54 – 55 #’s 14, 20, 26, 28, 30, 36, 42, 44, 50, 76, 77, 78
Work on this in class today and tonight for extra practice with fractional expressions.
We are going to review tomorrow, and have a small test on Monday!!!
Bell Work:
Simplify:1)
2)
Unit 1 Test
You need to know:
Exponents/Radicals (1.2)Add/Sub/Mult/Div Polynomials (1.3)Factoring (1.3)Fractional Expressions (1.4)
Review Problems
We have a Unit 1 test tomorrow on Lessons 1.2-1.4.
Pages 56-57, you should be able to do 13-84.
Today, work on the following in groups:
16, 20, 28, 30, 38, 40, 42, 66, 74, 76, 82, 84
For extra practice, you can try doing the odds at home to prepare!
Bonus for the Test!
A 5 inch wooden cube is painted blue. The cube is then cut into smaller 1 inch cube pieces. How many of the smaller 1 inch cubes have paint on 3 sides? 2 sides? 1 side? 0 sides?
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