Rational Expressions Algebra II Notes Mr. Heil Chapter 5

Rational Rational ExpressionsExpressions

Algebra II Notes Algebra II Notes

Mr. HeilMr. Heil

Chapter 5Chapter 5

Quotients of MonomialsQuotients of Monomials

TSWBAT simplify fractions TSWBAT simplify fractions using the 5 laws of using the 5 laws of

exponents. exponents.

Multiplying FractionsMultiplying Fractions

Let p, q, r, and s be real Let p, q, r, and s be real numbers with q ≠ 0 and s ≠ numbers with q ≠ 0 and s ≠ 0, then0, then

Example:Example:

qs

pr

s

r

q

p

35

12

75

43

7

4

5

3

Simplifying FractionsSimplifying Fractions

Let p, q, and r be real Let p, q, and r be real numbers with q ≠ 0 and r ≠ 0. numbers with q ≠ 0 and r ≠ 0. Then Then

Example: Example:

q

p

qr

pr

8

5

68

65

48

30

Laws of ExponentsLaws of Exponents

There are 5 Laws of Exponents.There are 5 Laws of Exponents.

All 5 laws start out with this All 5 laws start out with this generalized statement regarding generalized statement regarding the variables:the variables: Let Let aa and and bb be real be real numbers and numbers and mm and and nn be positive be positive integers and a ≠ 0 and b ≠ 0 integers and a ≠ 0 and b ≠ 0 when they are divisors.when they are divisors.

Laws of ExponentsLaws of Exponents

First LawFirst Lawaamm * a * ann== a am+nm+n

Example : xExample : x33*x*x55=x=x3+53+5=x=x88

Second LawSecond Law((abab))mm = = aammbbmm Example : (4x)Example : (4x)33=4=433xx33=64x=64x33

Third LawThird Law((aamm))nn = = aam*nm*n Example: (xExample: (x33))44=x=x3*43*4=x=x1212

Laws of ExponentsLaws of ExponentsFourth Law – Fourth Law –

Fifth Law - Fifth Law -

nmn

m

aa

a

m

mm

b

a

b

a

8222

2 3585

8

Examples:

933

6

2

2*323 ttt

Zero and Negative ExponentsZero and Negative Exponents

TSWBAT Simplify expressions TSWBAT Simplify expressions using zero and negative using zero and negative

exponents.exponents.

ExponentsExponents

Zero ExponentZero Exponent – If a ≠ 0, then – If a ≠ 0, then aa00=1.=1.

Ex. 2Ex. 200=1=1 xx00=1=1

Note: 0Note: 000 is not defined. is not defined.

ExponentsExponentsNegative ExponentNegative Exponent – If n is a – If n is a positive integer and a ≠ 0, positive integer and a ≠ 0, then then

Examples:Examples:

nn

aa

1

100

1

10

110

22

22 3

3c

c

xx

5

1)5( 1

Scientific NotationScientific Notation

TSWBAT use scientific and TSWBAT use scientific and decimal notation for numbers.decimal notation for numbers.


Scientific NotationScientific Notation – a method in – a method in which a number is expressed in which a number is expressed in the form m X 10the form m X 10nn, where , where 1≤m<10, and n is an integer.1≤m<10, and n is an integer.

Examples – Examples –

4006 in SN is 4.006 X 104006 in SN is 4.006 X 1033..

0.00203 in SN is 2.03 X 100.00203 in SN is 2.03 X 10--

33..


Decimal NotationDecimal Notation – The extended – The extended form of a number with all place form of a number with all place values. This is usually the values. This is usually the number you put into scientific number you put into scientific notation.notation.

Examples – 4006Examples – 4006

0.002030.00203


Significant DigitsSignificant Digits – any nonzero – any nonzero digit or any zero that has a digit or any zero that has a purpose other than placing the purpose other than placing the decimal point. decimal point.

Examples – The significant Examples – The significant digits are colored bright blue digits are colored bright blue 40064006, 0.00, 0.00203203

Rational Algebraic Rational Algebraic ExpressionsExpressions

TSWBAT: Simplify rational TSWBAT: Simplify rational algebraic expressions by factoring algebraic expressions by factoring

and the rules of simplifying and the rules of simplifying fractions.fractions.

Rational Algebraic ExpressionRational Algebraic Expression

Rational NumberRational Number – a number that – a number that can be expressed as a quotient can be expressed as a quotient of integers.of integers.

Rational Algebraic ExpressionRational Algebraic Expression – – is an expression that can be is an expression that can be expressed as a quotient of expressed as a quotient of polynomials.polynomials.

Example: Example:

4

22

2

x

xx

Rational FunctionRational Function

Rational FunctionRational Function – a function – a function that is defined by a simplified that is defined by a simplified rational expression in one rational expression in one variable.variable.

Example - Example -

xxx

xxxf

2

372)(

23

2

Simplifying Rational Algebraic Simplifying Rational Algebraic ExpressionsExpressions

Examples-Examples-

Product and Quotient of Rational Product and Quotient of Rational Algebraic ExpressionsAlgebraic Expressions

TSWBAT multiply and divide TSWBAT multiply and divide rational algebraic rational algebraic

expressions.expressions.

Products and Quotients of Products and Quotients of Rational ExpressionsRational Expressions

To find the product or quotient To find the product or quotient of two or more rational of two or more rational expressions we use the expressions we use the multiplication and division rules multiplication and division rules of fractions.of fractions.

Final answers should always be Final answers should always be expressed in simplest form. expressed in simplest form. Thus the rules of simplifying Thus the rules of simplifying fractions must be used.fractions must be used.

Multiplying FractionsMultiplying Fractions

Let p, q, r, and s be real Let p, q, r, and s be real numbers with q ≠ 0 and s ≠ numbers with q ≠ 0 and s ≠ 0, then0, then

Example:Example:

qs

pr

s

r

q

p

3

3

)2)(2(

)3)(2(

)3)(3(

)2(3

4

6

96

632

2

2

2

x

x

xx

xx

xx

xx

x

xx

xx

xx

Division of FractionsDivision of FractionsLet p, q, r, and s be real Let p, q, r, and s be real numbers with q ≠ 0, r ≠ 0, and numbers with q ≠ 0, r ≠ 0, and s ≠ 0, thens ≠ 0, then

Examples: Examples:

r

s

q

p

s

r

q

p

3

2

7

5

15

14

5

7

15

14

1

2

3

636 23

232

ax

y

xa

a

xy

xa

y

a

xy

Sums and Differences of Rational Sums and Differences of Rational Algebraic ExpressionsAlgebraic Expressions

TSWBAT add and subtract TSWBAT add and subtract rational algebraic rational algebraic

expressions. expressions.

Sums and Differences Sums and Differences of Rational Expressionsof Rational Expressions

If we have two or more rational If we have two or more rational expressions with the same expressions with the same denominator denominator c,c, the following is the following is true:true:

Example:Example:

c

ba

c

b

c

a

c

ba

c

b

c

a

2

12

2

)9()83(

2

9

2

83 22222

x

x

x

xx

x

x

x

x


If the denominators are not the same, the If the denominators are not the same, the following steps must be followed:following steps must be followed:

Step 1 – Find the LCD (Least Common Step 1 – Find the LCD (Least Common Denominator) – that is the LCM of the Denominator) – that is the LCM of the terms in the denominator.terms in the denominator.

Step 2 - Express each fraction as an Step 2 - Express each fraction as an equivalent fraction with the LCD as equivalent fraction with the LCD as the denominator.the denominator.

Step 3 – Add or Subtract and simplify.Step 3 – Add or Subtract and simplify.


Example:Example:

Find LCD which is 8abFind LCD which is 8ab22

Write equiv. fractionsWrite equiv. fractions

Add and simplifyAdd and simplify

28

3

2

1

bab

22 8

3

8

4

ab

a

ab

b

28

34

ab

ab


ExamplesExamples

Complex FractionsComplex Fractions

TSWBAT simplify complex TSWBAT simplify complex fractions by using the rules of fractions by using the rules of

multiplying and dividing fractions.multiplying and dividing fractions.

Complex FractionsComplex FractionsComplex Fraction –Complex Fraction – a fraction in which a fraction in which the numerator, denominator, or both the numerator, denominator, or both contain one or more fractions or contain one or more fractions or powers with negative exponents.powers with negative exponents.

Examples: Examples:

9

42

6

1

12

7

22

11

yx

yx


To simplify these fractions there To simplify these fractions there are two methods.are two methods.

1. First simplify the numerator, 1. First simplify the numerator, then the denominator separately then the denominator separately and then third divide the and then third divide the numerator by the denominator.numerator by the denominator.


Example:Example:

88

15

264

45

22

9

12

5

9

22

12

5

9

2212

5

9

41812

27

9

42

6

1

12

7


2. Multiply the numerator and 2. Multiply the numerator and denominator by the LCD of all denominator by the LCD of all the fractions appearing in the the fractions appearing in the numerator and denominator and numerator and denominator and then combine like terms.then combine like terms.


Example:Example:

88

15

1672

621

369

42

366

1

12

7

9

42

6

1

12

7

Fractional Coefficient Fractional Coefficient EquationsEquations

TSWBAT solve equations and TSWBAT solve equations and inequalities involving inequalities involving fractional coefficients.fractional coefficients.

Fractional CoefficientsFractional Coefficients

Fractional CoefficientsFractional Coefficients - To solve - To solve an equation or inequality with an equation or inequality with fractional coefficients we fractional coefficients we multiply both sides by the LCD of multiply both sides by the LCD of the fraction to turn the fraction the fraction to turn the fraction into a normal equation or into a normal equation or inequality.inequality.

Fractional CoefficientsFractional CoefficientsExample:Example:

5

3

3

1035

0130)35)(13(

034153415

10

1

15

230

230

10

1

15

2

222

22

xorxxor

xxx

xxxx

xxxx

Fractional EquationsFractional Equations

TSWBAT solve and use TSWBAT solve and use fractional equations and word fractional equations and word

problems.problems.


Fractional EquationFractional Equation - An - An equation in which a variable equation in which a variable occurs in the denominator.occurs in the denominator.

Example:Example:

5

42

107

32

x

x

xx


When solving a Fractional When solving a Fractional Equation by multiplying both Equation by multiplying both sides by the LCD the new sides by the LCD the new equation may not be equivalent equation may not be equivalent to the original. We must check to the original. We must check the answers at the end.the answers at the end.

Fractional EquationsFractional EquationsExample SolveExample Solve

Find the LCD: Find the LCD:

The LCD is: (x-2)(x-5) -> so:The LCD is: (x-2)(x-5) -> so:

5

42

107

32

x

x

xx

5

4

1

2

)5)(2(

3

x

x

xx


)2)(4()5)(2(23

)5)(2(5

4

2107

3)5)(2(

2

xxxx

xxx

x

xxxx


53

050)3(

0)5)(3(

0158

8623142

86201423

2

22

22

xorx

xorx

xx

xx

xxxx

xxxx


Then we need to check each Then we need to check each solution -> when we check x=5 solution -> when we check x=5 we find we have a zero we find we have a zero denominator which is not denominator which is not possible therefore x=5 is not a possible therefore x=5 is not a solution and only x=3 is correct.solution and only x=3 is correct.


Extraneous rootExtraneous root – a root of the – a root of the transformed equation that is not transformed equation that is not a root of the original equation.a root of the original equation.

Example: In the equation above Example: In the equation above 5 is an extraneous root as it 5 is an extraneous root as it solves the transformed equation solves the transformed equation but not the original.but not the original.

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Rational Expressions Algebra II Notes Mr. Heil Chapter 5