Upload
others
View
16
Download
0
Embed Size (px)
Citation preview
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 1
• Expressions
• Equations
• Inequalities
UNIT 3: RATIONAL FUNCTIONS
Lesson #1: Simplifying,
Multiplying & Dividing
Rational Expressions
What is a rational expression?
Recall: rational numbers can
be expressed as fractions.
A rational expression is a quotient (fraction) of two
polynomials in the form:
Any value of � that makes the
denominator equal zero is an
undefined value.
� � �����
����where ����
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 2
Undefined ValuesHow can we determine
undefined values given
a rational expression?
Example: � � �� ��
��
� Set the denominator
equal to zero.
� Solve the equation.
� Apply the zero
product property!
� � � 2 � 0
� � 0 � � 2 � 0
� � �2 � These values make the
expression undefined!
Undefined ValuesDetermine the value(s) of � that
make the expression undefined.
Ex 1) ��
����
Ex2) ��� ���
��� ���
� Set denominator
equal to zero.
� Solve the equation.
How?How?How?How?⟹ Factor! or
�� � �� �
⟹ Square root
property!
�� � ���� � � ��
� � ��
�� � � � �� � � Solve the
equation.
How?How?How?How?⟹ Factor!
�� � !� � � " �
� � ! � � � " �
� � "� � �!
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 3
Simplifying Rational Expressions
Recall, how are fractions simplified?
8
12
� · �
� · "
�
"
Simplifying fractions involves
the process of division.(WE DO NOT CANCEL!)
We are looking for
FACTORS of the
numerator and
denominator.
Remember: Anything
divided by itself is �.
Simplifying Rational ExpressionsSimplify each expression for all values
of � for which the expression is defined.
Ex 1) ��
����
*To simplify, the numerator & denominator must have a common factor. *Common terms cannot be simplified.
We are looking for FACTORS of the numerator and denominator.
�� � �
�� � ��� � ��
�
�� � �
Then we divide the common
factors.(WE DO NOT CANCEL!)
Remember: Anything
divided by itself is �.
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 4
Simplifying Rational ExpressionsSimplify each expression for all values
of � for which the expression is defined.
Ex 2) ��& ���
'��(
�� � ��� � "�
�� � "���� � "� � )��
� �
��� � "� � )�
How?How?How?How? ⟹ Factor!
QUICK REVIEW
Simplify:
(a)���
����
(b) ��(
(���
(c) ��
�� �
���
���
���
�
� � �
�� � ��
� � �
�� � � ��
Rewrite! Factor! Divide!
��
� � �
� � �� ��
Therefore:
“cousins”“cousins”“cousins”“cousins”Commutative
Property is not true for subtraction
They are “similar” [“cousins”],
therefore when we divide we get -1.
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 5
Simplifying Rational ExpressionsSimplify each expression for all values of
� for which the expression is defined.
Ex 3) � ��( ��
*� �
⟹��� � ���� � "�
�" � ���" � ��Sidework for numerator:
��� � +� � "
��� � �� � � � "
���� � "� � ��� � "�
��� � ���� � "�
���� � ��
" � �� �
��� � ��
" � �
⟹ Factor!��
Multiplying Rational Expressions
Recall, how are fractions multiplied?
5
12·10
25
Option 1) Multiply numerators
& denominators. Simplify.
"
· ���
�
DO NOT CROSS
MULTIPLY!!!!!!
Option 2) Factor, simplify, multiply
numerators & denominators.
� · �· · �
· �
�
We will use Option #2 for rational expressions.
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 6
Multiplying Rational ExpressionsWrite each expression in simplest form as
a single fraction for all defined values of �.
Ex 1) 8 � 4�
3� � 9·
�� � 9
�� � � � 6
⟹��� � ��
"�� � "�·�� � "��� � "�
�� � "��� � ��
⟹ Factor!
��
��
"
⟹��� � ��
·
⟹��� � ��
"�� � "�·
⟹
��� � ��
"�� � "�·�� � "��� � "�
Multiplying Rational Expressions
Write each expression in simplest form as a
single fraction for all defined values of �.
Ex 2) 2� � 23 � 63�
2� � 93�·22 � 63
2 � 23
⟹�4 � "5� 4 � �5
�4 � "5� 4 � "5·��4 � "5�
4 � �5⟹
�4 � "5� 4 � �5
�4 � "5� 4 � "5·
�
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 7
Dividing Rational Expressions
Recall, how are fractions divided?
6
216
2
27
Division is the inverse of
multiplication.Multiply by the reciprocal
�
��·�+
��" · �
+ · "·) · "
���+
+“KKKKeep CCCChange FFFFlip”
Dividing Rational ExpressionsWrite each expression in simplest form as
a single fraction for all defined values of �.
Ex 1) ��
6
���
'
� � �
�·
�"
�� � �
Multiply by the reciprocal Factor!
� � �
�·
� ⋅ ��
� � � � � �
Divide & Simplify!
��
� � �
Can this be
simplified
further?
NO!NO!NO!NO!There are NO
common factors!
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 8
Dividing Rational ExpressionsWrite each expression in simplest form as a
single fraction for all defined values of �.
Ex 2) ���
���� 6
�� �
��� �
� � "�
�� � ��·"� � ��
� � ��
⇒" � � �
� � � �·
⇒" � � �
� � � �·� � � � � �
� � � � � �⇒"
�·� � �
� � �
⇒" � � �
� � � �
More ExamplesWrite each expression in simplest form as
a single fraction for all defined values of �.
Ex 3) �� � 25
2� � 12·�� � 8� � 12
4� � 206�� � 7� � 10
8�
�� � �
�� � ��·�� � !� � ��
�� � �⋅
!�
�� � +� � �
� � � �
� � � �·� � � � � �
� � � ⋅
!�
� � � � �
� �!�
� ⋅ �
Alg2/Trig CC Unit 3
Lesson #1: Simplifying, Multiplying,
& Dividing Rational Expressions 9
More ExamplesWrite each expression in simplest form as
a single fraction for all defined values of �.
Ex 4) 2� � 1
32 � 9·2� � 82 � 15
42 � 462� � 62 � 5
62�
4� � �
"4 � )·4� � !4 � �
�4 � �·
�4�
4� � �4 �
4 � � 4 � �
" 4 � "·4 � " 4 �
� 4 � �·
�4�
4 � 4 � �
⇒" ⋅ � ⋅ 4�
" · � ⋅ ��4�
�