Rational Expressions

Student will be able to simplify rational expressionsAnd identify what values make the expression Undefined .

a.16

Simplifying Rational Expressions

The objective is to be able to simplify a rational expression-These are already!

5

2x

3

92

x

x

Undefined denominators Ignore the numerator Set the denominator = to zero and

solve

Undefined denominators-ex.

What value(s) would make these undefined

5

2x

3

92

x

x

Undefined denominators-ex.

What value(s) would make these undefined

5

2x

3

92

x

x X+2=0 x2 – 9 = 0X=-2

x+ 3 = 0 x – 3 = 0x = -3 x = 3

Try these: For what value of a are these

undefined:

1.5a2

4a

2.2

2x + 3

Answers: 1. 4a = 0 4 4 a = 0 2. 3a+2 = 0 -2 -2 3a = -2 3 3 a = -2/3

This is not reduced:

5c

10d

We do not have to factor monomial terms….

The greatest common factor is 5…divide it out both parts….

5c

10d=

1c2d

Try these:

Cancel all common factors….

1.1

2a22.

6x

7y3.

7

8

Answers:

Polynomial – The sum or difference of monomials.

Rational expression – A fraction whose numerator and denominator are polynomials.

Domain of a rational expression – the set of all real numbers except those for which the denominator is zero.

Reduced form – a rational expression in which the numerator and denominator have no factors in common.

Divide out the common factors

Factor the numerator and denominator and then divide the common factors

Dividing Out Common Factors

Step 1 – Identify any factors which are common to both the numerator and the denominator.

5

5 7

x

x( )The numerator and denominator have a common factor.

The common factor is the five.

Dividing Out Common FactorsStep 2 – Divide out the common factors.

The fives can be divided since 5/5 = 1

The x remains in the numerator.

The (x-7) remains in the denominator

5

5 7

x

x( ) x

x 7

x

x 7

Factoring the Numerator and Denominator

Factor the numerator.

Factor the denominator.

Divide out the common factors.

Write in simplified form.

3 9

1 2

2

3

x x

x

Factoring

Step 1: Look for common factors to both terms in the numerator.

3 9

1 2

2

3

x x

x

3 is a factor of both 3 and 9.

X is a factor of both x2 and x.Step 2: Factor the numerator.

3 9

1 2

2

3

x x

x

3 3

12 3

x x

x

( )

Factoring

Step 3: Look for common factors to the terms in the denominator and factor.

3 9

1 2

2

3

x x

x

The denominator only has one term. The 12 and x3 can be factored.

The 12 can be factored into 3 x 4.

The x3 can be written as x • x2.3 9

1 2

2

3

x x

x

3 3

3 4 2

x x

x x

( )

Divide and Simplify

Step 4: Divide out the common factors. In this case, the common factors divide to become 1.3 3

3 4 2

x x

x x

( )

Step 5: Write in simplified form.

x

x

3

4 2

You Try It

Simplify the following rational expressions.

19

2 4

2

2.

x yz

xyz

23

4 32.

a

a a

33 1 5

7 1 02.

x

x x

42 1 5

1 2

2

2. x x

x x

Reducing to -1

x + 77 + x

=1 butx−55 −x

=−1

x(x−3)(x+5)(3−x)

Reduce:

Answer:

x(x−3)(x+5)(3−x)

−xx + 5

-1

Student will be able to Multiply Rational Expressions and express in simplest form a2.a.16

5a2

16• 36b

5a

Do Now: Multiply:

1

2•69

Copy this:

Student will be able to Multiply Rational Expressions and express in simplest form a2.a.16

5a2

16• 36b

5a

Cross cancel common factors and then multiply acrossThe numerators and across the denominators:

5a2

16•36b5a

9

4

=9ab

4

Multiplying when factoring isnecessary!

x2

6•

3x+6x2 +2x

x2

6•3(x+2)x(x+2)

FACTOR:

Canceling step:

x2

6•3(x+2)x(x+2)

2

x

2=

Cancel top and bottom and on diagonals:

Multiply numerators, multiply denominators:

Ex:

12x2

5x +15•

x2 −93x2 +9x

Restrictions on Rational Expressions

For what value of x is undefined? x x

x

2 2 1 5

4 2 0

It is undefined for any value of “x” which makes the denominator zero.

4 2 0 0x

x 5

The restriction is that x cannot equal 5.

4x =20

YOU TRY ITWhat are the excluded values of the variables for the following rational expressions?

14 2

1 4

3 2

2 3.

y z

y z

23 6

2 1 2

2

. x

x

More complicated

What are the excluded values of the variables for the following rational expression. ? (undefined)

34 1 2

2 8

2

2.

c c

c c

Dividing Rationals

Student will be able to divide rational expressions and Express answer in simplest form.

Do Now: divide these fractions (remember that dividing isMultiplying by the reciprocal)

4

3÷

56

Answer

4

3÷

56

4

3•65

2

1 =8

5

Multiply by the reciprocal:a.k.a.:“Flip” and multiply

Algebraic Example:

x2 −25x2 +5x+6

÷5x−2510x+ 30

Note: after inverting, (“flipping”) the second expression,factor all four parts and follow multiplying rules

Algebraic Example:

x2 −25x2 +5x+6

•10x+ 305x−25

(x+5)(x−5)(x+2)(x+ 3)

•10(x+ 3)5(x−5)

=2(x+5)(x+2)

2

Example 2

(Completely factor the First numerator)

Example 2

(Completely factor the First numerator)

2(x−3)(x+2)4(x+2)

•9(x−3)

(x−3)(x−3)

=92

2

Do and hand in on exit card:

3a

a+ 3÷

5a2a+6

Adding/Subtracting Rational Expressions

1

2+

38

Do now: (remember common denominators)

Today, you will be able to add rational expressions by findingLeast common denominators…..

Adding/Subtracting Rational Expressions

1

2(4

4)+

38

=48

+38

=78

Algebraic examples:

2x

3+

5x6

(x+ 4)2

+(x−6)

3

Algebraic examples:

2x

3+

5x6

(x+ 4)2

+(x−6)

3

Lcd = 6

Lcd=6

Answers:2x

3• (

22)+

5x6

=4x6

+5x6

=9x6

=3x2

(x+ 4)2

+(x−6)

3=(

33)•

(x+ 4)2

+(22)•

(x−6)3

=

3x+126

+2x−12

6=

5x6

Distribute!

Subtracting-remember to distribute!

2x + 45

−4x−8

4

Subtracting-remember to distribute!

4

4• (

2x+ 45

)−55• (

4x−84

)=

8x+1620

+−20x+ 40

20=

−12x+5620

But this can be reduced!

Reducing:

−12x + 56

20=

4(−3x +14)

20=

(−3x +14)

5

Trickier denominators:

3

x + 3+

xx2 −9

Here we should factor the second denominator in order to findThe least common denominator…

Finding the lcd:

Which means (x+3)(x – 3) is the lcd so multiply the first Fraction by (x – 3)/(x – 3)

Answer:

3

x + 3+

xx2 −9

3x+ 3

•(x−3)(x−3)

+x

(x+ 3)(x−3)=

3x−9+ x(x+ 3)(x−3)

=4x−9x2 −9

Not reducable!

Next example:

6

x +1−

2x

Solution:

x

x•

6(x+1)

−2x•(x+1)(x+1)

=

6xx(x+1)

+−2x−2x(x+1)

=

4x−2x(x+1)

Try this-(factor to find lcd)

−24

x2 − 2x −15+

3

x − 5

This one will need to be reduced at the end….

Answer lcd = (x-5)(x+3):

−24

x2 − 2x −15+

3

x − 5=

−24

(x − 5)(x + 3)+

3

(x − 5)•

(x + 3)

(x + 3)

Answer:−24

x2 − 2x −15+

3

x − 5=

−24

(x − 5)(x + 3)+

3

(x − 5)•

(x + 3)

(x + 3)=

−24 + 3x + 9

(x − 5)(x + 3)=

3x −15

(x − 5)(x + 3)=

3(x − 5)

(x − 5)(x + 3)=

3

x + 3

Complex Fractions a.17

Student will be able to simplify complex fractions byMultiplying each term by the least common denominator and Simpifying if necessary.

Do Now - Divide:

1

x−1÷

1x2 −1

A fraction over another fraction

1

x−1÷

1x2 −1

=1

x−1•(x+1)(x−1)

1=x+1

Now think of it this way: This is called a complex fraction.

1

x−11

x2 −1

We flip the bottom and multiply, justLike when we divided.

Fractions within a fraction:

1

x+ 1

x2

2x+ x

Step 1-find the lcd of all 4 termsStep 2-multiply each term by the lcd/1

Fractions within a fraction:

1

x• x2

1+ 1

x2 • x2

12x• x2

1+ x

1• x2

1

Step 1-find the lcd of all 4 termsStep 2-multiply each term by the lcd/1

Lcd – x2

x

x=x +1

2x + x3

1

1

Example:

ab+ 1b

ab

− ab2

Solution: lcd = b2

ab

1• b2

1+ 1

b• b2

1ab

• b2

1− a

b2 • b2

1

Solution: lcd = b2

ab

1• b2

1+ 1

b• b2

1ab

• b2

1− a

b2 • b2

1

=ab3 +b

ab − a

b

b

Next example:

2 − 4ab

2a

+ 2b

lcd: ab2

1• ab

1− 4

ab• ab

12a• ab

1+ 2

b• ab

1

=2ab−42b+2aBut this one needs to be reduced!

lcd: ab

=2ab − 4

2b+ 2a

=2(ab − 2)

2(b+ a)

=ab − 2

b+ a

Solving Rational equations:

Do now: page 60 # 11,12

Solving rational equations using the lcd method:

1

a+

13

=32a

−1

1. Find the lcd of all terms2. Multiply each term by the lcd3. Solve the equation

How is this different than the ones you just solved?

STEPS:

Solution:

1

a•6a1

+13•6a1

=32a

•6a1

−11•6a1

2 3

6 + 2a = 9 – 6a

Look, we eliminated denominators!

+6a +6a6 +2a=9 −6a

6 +8a=9

8a=3

a=38

-6 -6

____ ___ 8 8

Example:

2 +3x

=102x

Lcd=2x

4x + 6 = 10 4x = 4 x = 1

Try this:

a

a+2=

3a

+4

a2 +2a

Lcd=a(a+2)

a

a+2•

a(a+2)1

=3a•

a(a+2)1

+4

a2 +2a•

a(a+2)1

a2=3a+6+4a2=3a+10a2-3a-10=0(a-5)(a+2)=0 a=5, a=-2

Extraneous roots:

Sometimes, when we check roots in the originalEquation, we arrive at an undefined denominator.These are called extraneous roots.

Check the roots in the previous problemWhich one is extraneous? Why?

Review

Do Now: Solve for x:

4

3x−2−

73x+2

=1

9x2 −4

Students will review rational expressions and equations

Review Rationals-index cardreview problems

Multiply and express in simplest form:

x2 −x3

•6

x2 −1

For what value of x is this undefined?4x

4 −x

1.

2.

Review RationalsAdd or subtract and express in simplest form:

3

x−2+

4x2 −4

1

x2 −1

1+ 1x

a

a−1−

aa+1

3. 4.

5.Express this complex fraction in simplest form:

Solving:

1

2+

5x−2

=36.

AnswersAdd corrected problems to index card for folder…

1.2x

x +12.4

3.3x+10x2 −4

4.2a

a2 −1

5.x

x3 + x2 −x−16.4

Finding the LCD

It is sometimes necessary to factor the denominators!

1

2a+2+

1a2 −1

x

x2 −4x+ 3−

xx2 +2x−3

b2

b−3+

93−b