UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying,...

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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

Transcript of UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying,...

Page 1: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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 ����

Page 2: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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!

�� � !� � � " �

� � ! � � � " �

� � "� � �!

Page 3: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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 �.

Page 4: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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.

Page 5: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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.

Page 6: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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·

Page 7: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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!

Page 8: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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�

�� � �

�� � ��·�� � !� � ��

�� � �⋅

!�

�� � +� � �

� � � �

� � � �·� � � � � �

� � � ⋅

!�

� � � � �

� �!�

� ⋅ �

Page 9: UNIT 3: RATIONAL FUNCTIONS Lesson 1 Simplifying...Alg2/Trig CC Unit 3 Lesson #1: Simplifying, Multiplying, & Dividing Rational Expressions 7 Dividing Rational Expressions Recall, how

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�