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Rational Expressions

Student will be able to simplify rational

expressions And 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!

Undefined denominators

- Ignore the numerator
- Set the denominator to zero and solve

Undefined denominators-ex.

- What value(s) would make these undefined

Undefined denominators-ex.

- What value(s) would make these undefined

X20 x2 9 0 X-2 x 3 0 x 3

0 x -3 x 3

Try these

- For what value of a are these undefined

Answers

- 1. 4a 0
- 4 4
- a 0
- 2. 3a2 0
- -2 -2
- 3a -2
- 3 3
- a -2/3

This is not reduced

We do not have to factor monomial terms.

The greatest common factor is 5divide it out

both parts.

Try these

Cancel all common factors.

Answers

Vocabulary

- 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.

Simplifying Rational Expressions

- 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.

- The numerator and denominator have a common

factor. - The common factor is the five.

Dividing Out Common Factors

- Step 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

Factoring the Numerator and Denominator

- Factor the numerator.
- Factor the denominator.
- Divide out the common factors.
- Write in simplified form.

Factoring

Step 1 Look for common factors to both terms in

the numerator.

- 3 is a factor of both 3 and 9.
- X is a factor of both x2 and x.

Step 2 Factor the numerator.

3

3

x

x

(

)

3

12

x

Factoring

Step 3 Look for common factors to the terms in

the denominator and factor.

- 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

3

x

x

(

)

2

3

4

x

x

Divide and Simplify

Step 4 Divide out the common factors. In this

case, the common factors divide to become 1.

Step 5 Write in simplified form.

x

3

2

x

4

You Try It

Simplify the following rational expressions.

Problem 1

Divide out the common factors.

Write in simplified form.

Problem 2

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

Problem 3

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

Problem 4

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

Reducing to -1

Reduce

Answer

-1

Student will be able to Multiply Rational

Expressions and express in simplest form a2.a.16

Do Now Multiply

Copy this

Student will be able to Multiply Rational

Expressions and express in simplest form a2.a.16

Cross cancel common factors and then multiply

across The numerators and across the

denominators

9

4

Multiplying when factoring is necessary!

FACTOR

Canceling step

Cancel top and bottom and on diagonals

2

Multiply numerators, multiply denominators

Ex

Restrictions on Rational Expressions

It is undefined for any value of x which makes

the denominator zero.

The restriction is that x cannot equal 5.

YOU TRY IT

What are the excluded values of the variables for

the following rational expressions?

Problem 1

Solution y ? 0 z ? 0

Problem 2

Solution 2x - 12 0 2x - 12 12 0

12 2x 12 2x ? 2 12

? 2 x 6

ANSWER x ? 6

More complicated

What are the excluded values of the variables for

the following rational expression. ? (undefined)

Problem 3

Solution C2 2C - 8 0 (C-2)(C4) 0 C-2

0 or C 4 0 C 2

or C -4

Answer C ? 2 C ? -4

Dividing Rationals

Student will be able to divide rational

expressions and Express answer in equivalent

simplest form. Do Now divide these fractions

(remember that dividing is Multiplying by the

reciprocal)

Answer

Multiply by the reciprocal a.k.a. Flip and

multiply

2

1

Algebraic Example

Note after inverting, (flipping) the second

expression, factor all four parts and follow

multiplying rules

Algebraic Example

2

Example 2

(Completely factor the First numerator)

Example 2

(Completely factor the First numerator)

2

When division looks different

Example when one is not a fraction..

Do and hand in on exit card

Adding/Subtracting Rational Expressions

Do now (remember common denominators)

Today, you will be able to add rational

expressions by finding Least common

denominators..

Adding/Subtracting Rational Expressions

Algebraic examples

Algebraic examples

Lcd 6

Lcd6

Answers

Distribute!

Subtracting-remember to distribute!

Subtracting-remember to distribute!

But this can be reduced!

Reducing

Trickier denominators

Here we should factor the second denominator in

order to find The least common denominator

Finding the lcd

Which means (x3)(x 3) is the lcd so multiply

the first Fraction by (x 3)/(x 3)

Answer

Not reducable!

Next example

Solution

Try this-(factor to find lcd)

This one will need to be reduced at the end.

Answer lcd (x-5)(x3)

Answer

Complex Fractions a.17

Student will be able to simplify complex

fractions by Multiplying each term by the least

common denominator and Simpifying if necessary.

Do Now - Divide

A fraction over another fraction

Now think of it this way This is called a

complex fraction.

We flip the bottom and multiply, just Like when

we divided.

Fractions within a fraction

Step 1-find the lcd of all 4 terms Step

2-multiply each term by the lcd/1

Fractions within a fraction

Step 1-find the lcd of all 4 terms Step

2-multiply each term by the lcd/1

x

1

Lcd x2

1

x

Example

Solution lcd b2

Solution lcd b2

b

b

Next example

lcd ab

But this one needs to be reduced!

lcd ab

Exit ticket Simplify

Solving Rational equations

Do now page 60 11,12

Students will solve rational equations,

exploring two methods that develop the skills

learned for adding and multiplying rational

expressions, and monitoring for the creation of

extraneous solutions. This will be evidenced by

an exit problem.

Solving rational equations using the lcd method

How is this different than the ones you just

solved?

- Find the lcd of all terms
- Multiply each term by the lcd
- Solve the equation

STEPS

Solution

2

3

6 2a 9 6a

Look, we eliminated denominators!

6a 6a

-6

-6

____ ___ 8 8

Example

Lcd2x

4x 6 10 4x 4 x 1

Try this

Lcda(a2)

a23a64 a23a10 a2-3a-100 (a-5)(a2)0 a5,

a-2

Extraneous roots

Sometimes, when we check roots in the

original Equation, we arrive at an undefined

denominator. These are called extraneous

roots. Check the roots in the previous

problem Which one is extraneous? Why?

Extraneous roots

Sometimes, when we check roots in the

original Equation, we arrive at an undefined

denominator. These are called extraneous

roots. Check the roots in the previous

problem Which one is extraneous? Why? -2 is

extraneous because it made part of the equation

undefined, so the solution is 5.

Review

Today we will review simplifying rational

expressions and solving rational equations for

your test tomorrow.

Do Now Solve for x

Review Rationals-index card review problems

Multiply and express in simplest form

1.

For what value of x is this undefined?

2.

Review Rationals

Add or subtract and express in simplest form

3.

4.

5.

Express this complex fraction in simplest form

Solving

6.

Answers

Add corrected problems to index card for folder

Just Find the LCD

It is sometimes necessary to factor the

denominators!

Just Find the LCD

It is sometimes necessary to factor the

denominators!