Real Numbers and Algebraic Expressions

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Real Numbers and Algebraic Expressions
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The Basics About Sets
The set 1, 3, 5, 7, 9 has five elements.

• A set is a collection of objects whose contents
  can be clearly determined.

• The objects in a set are called the elements of
  the set.

• We use braces to indicate a set and commas to
  separate the elements of that set.

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• ________________
• ________________
• ________________
• ________________

4

The set of even counting numbers is a
________________ of the set of counting numbers,
since each element of the subset is also
contained in the set.

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Integers

• The set of integers

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Important Subsets of the Real Numbers
7

8

• Definition
• Rational Numbers

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The Real Numbers

• Rational numbers

Irrational numbers

Integers
Whole numbers
Natural numbers
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The Real Number Line
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Graphing on the Number Line

• Which numbers are plotted?

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CE

• Symbols
• Less than
• Less than or equal
• Greater than
• Greater than or equal
• Not equal

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Absolute Value

• Absolute value describes the distance from 0 on a
  real number line. If a represents a real number,
  the symbol a represents its absolute value,
  read the absolute value of a.

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Definition of Absolute Value

• The absolute value of x is given as follows

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Properties of Absolute Value
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CE

• Find the following -3 and 3.

Solution
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Homework

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• 25 29 odd

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Distance Between Two Points on the Real Number
Line

• If a and b are any two points on a real number
  line, then the distance between a and b is given
  by

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CE

• Find the distance between 5 and 3 on the real
  number line.

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CE

• Find the distance between 20 and 8 on the real
  number line.

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CE

• Review
• Evaluate

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CE

• Review
• Evaluate the expression for x 3 and y 2

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CE

• Find the distance between the following points.
  (use absolute value)
• -2 and 5

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Algebraic Expressions

• A combination of variables and numbers using the
  operations of addition, subtraction,
  multiplication, or division, as well as powers or
  roots, is called an algebraic expression.

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CE

• List 3 examples of algebraic expressions

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CE

• !

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The Order of Operations Agreement

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

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CE

• Evaluate the expression
• 5x 4 If x 3

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CE

• Evaluate the expression
• 5(x 2)-5 If x 3

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CE

• The algebraic expression 2.35x 179.5 describes
  the population of the United States, in millions,
  x years after 1980. Evaluate the expression when
  x 20. Describe what the answer means in
  practical terms.

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Properties of Real Numbers

• For all real numbers a,b, and c
• Closure Properties
• Addition Multiplication

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Continued

• Commutative Property
• Addition Multiplication

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Continued

• Associative Property
• Addition Multiplication

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Continued

• Distributive Property

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• Zero Product Property

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• Inverse Property
• a (-a) (-a) a 0

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SEE HANDOUT
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Properties of the Real Numbers
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Properties of the Real Numbers
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Properties of the Real Numbers
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Properties of Negatives

• Let a and b represent real numbers, variables, or
  algebraic expressions.
• (-1)a -a
• -(-a) a
• (-a)(b) -ab
• a(-b) -ab
• -(a b) -a - b
• -(a - b) -a b b a

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• State the name of the property illustrated
• X b b X
• (a b) h a ( b h)
• a(b d) ab ad

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CE

• Simplify 6(2x 4y) 10(4x 3y).

Solution 6(2x 4y) 10(4x 3y)
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HomeworkPage1131 45 odd49 55 odd59-63
odd81
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Section P.2

• Exponents and Scientific Notation

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Definition of Positive Exponents

• If n is a positive integer and b is any real
  number, then
• Where b is the base and n is the exponent.

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Rules of Exponents
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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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Definition

• If b is a real number not equal to zero, then

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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Definition

• If n is an integer and b is a real number not
  equal to zero, then

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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Homework

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• 1 39 odd

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Definition

• Scientific Notation
• Move the decimal to obtain a number greater than
  or equal to 1 and less than 10.
• Count the number of places you moved the decimal.

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Continued

• If you moved the decimal to the left the exponent
  is positive.
• If you moved the decimal to the right the
  exponent is negative.

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CE

• Write 1,575,000,000,000 in scientific notation.

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CE

• Write in scientific notation
• 3,450,000

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• Page 22
• 2 40 even
• 65,69,73,77

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P.3

• Radicals

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Definition of a Radical

• is a radical
• n is the index
• a is called the radicand.

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Definition of

• Let a be a real number and n a positive integer
  2.
• If a gt 0, and then for some positive integer
  x.

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Continued

• If a lt 0 and n is odd, then is the negative
  number x such that .
• If a lt 0 and n is even, then is not a real
  number

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CE

• List an example where n is odd and a is less than
  zero.

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Definition

• For a real number a and positive integer n,

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CE

• Evaluate the radical

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CE

• Evaluate the radical

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CE

• Evaluate the radical

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Definition of

• Let be a rational number with
• if b is a real number such that is defined,
  then

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CE

• Rewrite using fractional exponents.

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CE

• Rewrite using fractional exponents.

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CE

• Rewrite using fractional exponents.

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CE

• Rewrite using fractional exponents.

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CE

• Evaluate

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• 77, 79, 81

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Addition of Radicals

• In order to add or subtract radicals, they must
  have the same index and the same radicand.

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CE

• Add

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CE

• Add

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CE

• Add

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CE

• Add

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• To rationalize a denominator you must multiply
  both the numerator and the denominator by the
  radical that is currently in the denominator.

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CE

• Rationalize the denominator

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CE

• Rationalize

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Rationalize with two terms

• If the denominator has two terms
• Multiply the numerator and denominator by the
  conjugate.

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CE

• If the denominator contains
• A. multiply by

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CE

• If the denominator contains
• A. multiply by

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CE

• Rationalize the denominator

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CE

• Rationalize the denominator

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Note

• In order to multiply or divide radicals, the
  index must be the same.

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Rules for Radicals.

• b not zero

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CE

• Multiply

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CE

• Multiply

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CE

• Divide

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Today

• Review sections p1-p3
• Online practice test
• Take the test
• When you are finished print the answers you
  missed
• Figure out what you did
• See me after school today or Monday morning
  before school for help.

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Section p.4

• Operations with Polynomials

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Definition

• Monomial
• .
• ,

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Definition

• Polynomial

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Definition

• Binomial -

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Definition

• Trinomial -

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Definition

• Like terms

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Combining like terms

• What is the sum of

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What is the difference
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Adding Polynomials

• Add the coefficients of like terms.

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CE

• Add the following binomials

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CE

• Add the following polynomials

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Subtracting Polynomials

• Must distribute the (negative sign)

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Ce

• Subtract

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CE

• Subtract

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Multiplying Binomials

• F - First
• O - Outer
• I - Inner
• L - last

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CE

• Foil
• (x3)(x1)

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CE

• Foil
• (x1)(x2)

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CE

• Foil
• (x5)(-x1)

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CE

• Foil
• (-2x-3)(3x1)

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CE

• Foil
• (-2x-3)(3x1)

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Note

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CE

• Evaluate

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CE

• Evaluate

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Factoring Trinomials

• This is the opposite of foil.
• We are now going to work backwards.

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Factor out the greatest common factor

• Find the highest degree expression that divides
  each term of a polynomial.

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CE

• Factor out the GCF

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CE

• Factor out the GCF

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CE

• Factor by grouping

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CE Factor
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CE

• Factor

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CE

• Factor

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CE

• Factor

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CE

• Factor

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CE

• Factor

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CE

• Factor

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CE

• Factor

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• Page 57
• 1 7 odd
• 11,13
• 17 25 odd

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The Difference of Two Squares
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CE

• Factor

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CE

• Factor

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CE

• Factor

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CE

• Factor

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The Difference (Sum ) of Two Cubes
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CE

• Factor

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CE

• Factor

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CE

• Factor

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CE

• Factor

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CE
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CE

• Factor

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• 41,43
• 49,51

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• Worksheet in class and hw
• Quiz p4 - p5

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Section p.6

• Rational Expressions

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• Domain.
• Look for fractions
• You cannot divide by zero

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CE

• What number must be excluded from the domain?

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CE

• What number must be excluded from the domain?

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Simplifying Rational Expressions

• 1. Factor the numerator and denominator.
• 2. Cancel out common factors.

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CE Simplify

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CE Simplify

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Multiplying Rational Expressions

• 1. Factor all numerators and denominators
  completely.
• 2. Cancel common factors
• 3. Multiply the numerators and multiply the
  denominators.

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Rule
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CE

• Multiply

170
CE

• Multiply

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CEMultiply

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• Page 68
• 1,2
• 7-13 odd
• 15-21 odd

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Theorem

• If are rational
• expressions, then

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CE

• Add

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CE

• Subtract

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• When the denominators are different, you must
  find a common denominator.

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CE

• Add

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CE

• Add

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CE Add
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Dividing Rational Expressions

• Invert the divisor and multiply.

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CE

• Divide

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CE

• Divide

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• Page 68
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• 33- 39 odd
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Test Chapter P