module 1- rational alg expression 2

8/10/2019 Module 1- Rational Alg Expression 2

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Module 1Rational Algebraic Expressions

What this module is about

This module is about rational algebraic expressions, or simply rational expressions.As you go over the exercises, you will develop skills in: finding the domain of the givenrational expressions, finding the values that will make a rational expressions meaningless,finding numerical values for the given values of x, and simplifying rational algebraicexpressions. Treat the lessons with fun and take time to go back and review if you thinkyou are at a loss.

What you are expected to learn

This module is designed for you to:

1. Define and illustrate rational expressions. !ind the domain of a rational expression". !ind the values that will make rational expressions meaningless#. !ind the numerical values of rational expressions for the given values of the

variable$s%. &ecall how to factor polynomials'. (implify rational algebraic expressions

How much do you know

1. )n *"x**, the value of the variable that must be excluded is x + 1 a. b. + c. d. +

. The domain of the variable x in the rational expression in 1 is

a. -x & x / b. -x & x + / c. -x & x / d. -x & x +/

". !ind the value of the variable that must be excluded in m + #m 0 #. m0 m + '

#. ive the domain of the variable m in the rational expression m + #m 0 #. m0 m + '

2

1

2

1

2121


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%. !ind any value for variable p which "x + # is meaningless. %p + %

'. !ind the numerical value of a rational expression + %x 0 1 when x 2 .x

3. !ind the numerical value of a rational expression 4x

y * when x 2 +1 and y 2 ". +"x 0 y

4. &ename + 4y'5%x# in lowest term. 'y35"x%

6. !actor a 0 ac + ab + bc by grouping and then rename into lowest term. a+ b

17. (implify r + r + 7 . r0 r + "7

What you will do

8esson 1

Definition of &ational 9xpression and !inding the Domain of its ariable

Algebraic expressions containing variables that are written in fractional formof the form - p, ; are polynomials, the polynomial ; 7/ are called rationalexpressions.

Examples:a 0 b , y '< same as y '=, 4k"+ "k 0 k 0 ' , #6abc , *1 and 1".

a + b 1 #k0 %k 1b% w %

Domain of &ational 9xpressions:

The denominators of the above expressions should not be e;ual to 5ero because

any value assigned to a variable that results in a denominator of 5ero will make theexpression meaningless and must be excluded from the domain of the variable.

!or 3a, exclude b 2 7. The domain of the expression is the set ofb real numbers - b & b 7/ except 7.

!or 1; + #, exclude ; 2 + '. The domain of the variable ; is the set of

; 0 ' real numbers - ; & ; + '/ except + '.

q

p

2


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!or m + % , exclude n 2 and n 2 ".n+ %n 0 '

The factors of n+ %n 0 ' is


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9xclude the values for which a+ a + 1 2 7.

!actor the polynomial and set each factor e;ual to 7.

a+ a + 1 2 7%wxy* 6. * m + %m 0 ' * x


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?. ive the domain of the variable for each of the following algebraic expressions.

1. *1"b* '. *x


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Example:

1. !ind the numerical value of "x 0 ' from the given replacement set x 2 -1, , "/.x + #

)f x 2 1, replace x with 1.

"x 0 ' ""*3' + "x + x ' + "


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. * "x * 3. * x 0 1 * + #x 0 1 x+ 3x 0"

". 3x + x 4. **+ x * *

4x 4 0 x + x

#. * x** 6. * x 0 % * "x + 1 x0 "x +17

%. *


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8esson "

&ecalling Bow to !actor Colynomials

?efore you proceed to simplifying rational expressions, you must review first some

concepts that will make your task easier.

Lets have another look at factoring polynomials and get connected!

Greatest Common Factors of Polynomials

A polynomial, specifically monomial, is written in factored form when it is expressedas the product of prime numbers and variables where no variables has an exponentgreater than 1.

Examples

1. !actor #%ab#.

#%ab#2 % 6 a a b b b b

The @! of two or more monomials is the product of their common factors, wheneach monomial is expressed as a product of prime factors.

. !ind the @! of 1a"bc#and "7abc".

1a"bc#2 " a a a b c c c c

"7abc"2 " a b c c c % b

@!2 " a b c c c 2 'abc"

The @! of 1a"bc#and "7abc"is 'abc".

Factoring Using the Distributive Property

". se the distributive property to factor #x"y50 "'xy5".

!irst find the @! for #x"y50 "'xy5". ote that 1 is the largest number that will

divide evenly into # and "', the numerical coefficients of the polynomials. Also, xy5 isthe largest factor of x"y5and xy5".

#x"y50 "'xy5"2 1xy5


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#. !actor #a#b+ 'a"b"0 14ab#.

The @! of the numerical part is '. To find the variable part, write each variable theleastnumber of times it appears in any term. Thus ais the least power of a that appears,while b is the least power of b that appears. sing this, you can see that 'a b is thegreatest common factor, and so you have

#a#b+ 'a"b"0 14ab#2 'ab


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3. !actor "pw + 1w 0 %p + "%.

"pw + 1w 0 %p + "% 2


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Hhen using the !I)8 method, take note of the product of the coefficients of the firstand last terms, 14 and 6. otice that it is the same as the product of the two terms, 14 and6, whose sum is the coefficient of the middle term. Eou can use this pattern to factor;uadratic trinomials, such as 'x0 "x 0 7.

17. !actor 'x0 "x 0 7.

'x0 "x 0 7 The product of ' and 7 is 17.'x0 < ** 0 ** =x 0 7 Eou need to find two integers whose

product is 17 and whose sum is ".

Bow about using the guess>and>check strategy to find these numbers.

Factors of 120 Sum of Factors

1, 17 1 0 17 2 11 no

, '7 0 '7 2 ' no", #7 " 0 #7 2 #" no#, "7 # 0 "7 2 "# no%, # % 0 # 2 6 no', 7 ' 0 7 2 ' no4,1% 4 0 1% 2 " yes

'x0


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c. The only factors that gives the middle term is 03x is


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Factoring the Sum of Two Cubes

The sum of two cubes a"0 b" can be factored by writinga"0 b"2


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3 "


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Try this out

A. Hrite each expression in lowest terms.

1. *1k '. *"y + "y *

'k


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#. * n *+ p 6. *xy + yw 0 x5 + 5w * p + n xy 0 yw 0 5x 0 5w

%. *r + r + 7 * 17. ac 0 ad + bc + bd *

r

0 r + "7 ab 0 ac + b

+ bc

Lets ummari!e

&ational expression is the ;uotient of two polynomials with denominator note;ual to 5ero. &ational expression is any element of the set - p, ; are polynomials,the polynomial ; 7/.

Any value$s assigned to a variable that results in a denominator of 5ero will make arational expression meaningless.

To find the numerical values of rational expressions you will get the values of x fromthe replacement set. The set of numbers from which replacements for a variable may bechosen is called a replacement set.

!actoring is the process of writing an indicated sum as a product of factors.

!actoring a polynomial or finding the factored form of a polynomial means to find itscompletely factored form.

To find the variable partof the polynomial you are factoring, write each variable theleastnumber of times it appears in any terms of the polynomial.

The fundamental property of rational expressions permits us to write a rationalexpression in lowest terms, in which numerator and denominator have no common factorother than 1.

The ;uotient of two expressions that are exactly opposite in sign is +1.

What have you learned

1. )n *'x 0 x** , the value of the variable that must be excluded are ******. #x+ 1'

. ive the domain of the variable x in the rational expression **a 0 1**. a0 3a 0 1

q

p

17


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". !ind the value of the variable that must be excluded in m3n#p4 * . >14m1np'

6. (implify *y# + 1"y 0 "'* and then rename into lowest term. y0 %y 0 '

17. (implify *x# + 1' * . x#+ 4x0 1'

18


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"ey to correction

%o& much o you 'no&

1. x + 1 2 7 '. + %6 > 1x 2 1 letter c


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1. Domain 2 -b & b 7/ '. Domain 2 -x, y & x 7N y >/. Domain 2 -y, 5 & y, 5 7/ 3. Domain 2 -p & p >', %/

". Domain 2 -m & m %/ 4. Domain 2 -c & c "/#. Domain 2 -x, w & x 7N x ", >"/ 6. Domain 2 -m & m 7, '/

%. Domain 2 -y & y #, >#/ 17. Domain 2 -s & s , >/

C*1. x 2 7 '. 17s;uare root of negative number. o value

#esson ,

)*

x - . , x - / x - ,

1 #= + % >4 > % >1" 11 '= >1 >1 1

##3 3

x - . , x - / x - ,

" 3= + = > 1 " 4= >1' 4

31

** ' 0 6 "

F >4


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' ***> 0 4**** ' "=+ #= 0 1# 3

***7 0 4**** 4 ##0% >1=0"= >17 >1 1

**117 7 meaningless

17 **"= + 3*** >'>3 >1"= > "= > 1 1

*" 3 " 1 " >

**"41


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C*1. >x+ 4 2 7 '. x+ 4x 0 3 2 7 >x2 4 # o real number value.. >#x 0 1 2 7 3. x+ #x 0 " 2 7 >#x 2 >1 < x + "= x0 % 2 7 4x"2 1 > x2 + % 2 x2 % x 2 1 x 2 %

x 2 ot an integer x 2 + % not a positive integer

#. "x+ 1 2 7 6. x0 14x 0 41 2 7 "x2 1


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2


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". '


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. * m + #*** "N #

#. x+ x + # 2 7


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%. #x+ '# 2 7 #x2 '# x 2 #N + #

'. "= + %"= 0 " * m3n#p4 * **np* >14m1np' 14m%

6. *y # + 1"y 0 "'* *

module 1- rational alg expression 2

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