Simplifying Rational Expressions – Part II. This PowerPoint contains notes, examples and practice...

Simplifying Rational Expressions – Part II

This PowerPoint contains notes, examples and practice problems for you to complete. It is important that you slowly work through this

PowerPoint and not just move through it quickly. This is your chance to practice this material, so you must work through it and learn it!

Remember that, as you work through this PowerPoint, do not go onto the next part until you are certain that you understand what is being

presented.

Also, do not look at any answers until you have tried the problem yourself.

If you have any questions, ask your teacher!

Before we begin looking at simplifying rational expressions, we need to review a few ideas about numbers and fractions. So, we need to review the following vocabulary words. See if you can remember what each means before you look at the definition. They are:

Factor:

Multiple:

These terms are all important when we look at simplifying fractions.

Prime Number:

factor is a portion of a quantity, usually an integer or polynomial that, when multiplied by other factors, gives the entire quantity. The determination of factors is called factorization or factoring.

A multiple of a number x is any quantity y = nx with n an integer. If x and y are integers, then x is called a factor of y.

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p > 1 that has no positive integer factors other than 1 and p itself.

So, if I have two numbers, I can look for its least common multiple. That is we not only want a new number which can be divided by both of our other numbers but also the smallest number that can be divided by both numbers.

To do this, we can look at the factors of each number. If there is a factor that is in common between both numbers, we only need to use it once.

We know that 6 has factors 2 and 3 while 10 has factors 2 and 5. So, we know that the least common multiple between 6 and 10 will have factors of 2, 3 and 5, which is 30.

Now, 60 is a multiple of 6 and 10 but it is not the smallest multiple. We are always looking for the smallest multiple.

Here is an example:

What is the least common multiple between the numbers 6 and 10?

Notice that 30 can be divided evenly by 6 and 10. Because this is true, 30 is a multiple of 6 and 10.

Therefore, our answer is 30.

We know that 4 has factors 2 and 2 while 6 has factors 2 and 3. So, we know that the least common multiple between 4 and 6 will have factors of 2, 2 and 3, which is 12. We need two factors of 2 but not 3!

Now, 24 and 36 and 48, for example, are all multiples of 4 and 6 but they are not the smallest multiples. We are always looking for the smallest multiple.

Here is one more example:

What is the least common multiple between the numbers 4 and 6?

Notice that 12 can be divided evenly by 4 and 6. Because this is true, 12 is a multiple of 4 and 6.

Therefore, our answer is 12.

It is now your turn, find the least common multiple between the following numbers. Do NOT look at the answers until you have tried them!

6 and 8 10 and 4

6 and 9 8 and 12

Answer: Answer:

Answer: Answer:

24

18

20

24

This is probably easy for many of you but important to understand. If you are not sure, ask! You need to know how to find the least common multiple.

15 and 20 10 and 12

Answer: Answer:

60 60

Now, the same idea can be applied to an Algebraic Expression! Remember an expression has both numbers and letters, but we can FACTORIZE it and figure out what are the factors of the expression.

Remember, that because our expression has both numbers and letters in it, our factors should have both numbers and letters in it.

Here is an example, what is the least common multiple between the expressions below: 2 26 and 2x x x x So, we are looking for an algebraic expression which contain the least factors of both expressions.

Guess what, we are factorizing again! We have several methods of factorizing that we can use and MUST understand. When we factorize, we have:

Notice, we will need only one x, because it is in common with both expressions but we also need (x + 6) and (x + 2). Therefore:

( 6)( 2)x x x is our least common multiple!

2 26 ( 6) and 2 ( 2)x x x x x x x x

Here is an example, what is the least common multiple between the expressions below:

2 21 and 3 2x x x

Remember, we are looking for an algebraic expression which contain the least factors of both expressions.

When we factorize, we have:

Notice, we will need only one (x + 1), because it is in common with both expressions but we also need (x + 2) and (x - 1). Therefore:

( 1)( 1)( 2)x x x is our least common multiple!

2 21 ( 1)( 1) and 3 2 ( 1)( 2)x x x x x x x

It is now your turn, find the least common multiple between the following numbers. Do NOT look at the answers until you have tried them!

24 8 and 4x x 2 22 and 4 3x x x x

2 22 and 3x x x x 2 24 4 and 6 12x x x x

Answer: Answer:

Answer:Answer:

4( 2)( 2)x x

( 2)( 3)x x x

( 1)( 2)( 3)x x x

12 ( 1)( 2)x x x

The order in our answer doesn’t matter, as long as we have all of the factors! The key part to remember is that you need to factorize!

2 315 and 10x x 3 2 3 22 and 2x x x x x Answer:

Answer:

330x2 (2 1)( 1)x x x

Here is some more practice, find the least common multiple between the following numbers. Do NOT look at the answers until you have tried them and don’t move ahead until you are confident you know what to do!

2 23 6 and 2x x x x 2 29 and 4 3x x x

2 22 5 2 and 2 7 3x x x x 2 22 2 and 3 9x x x x

Answer: Answer:

Answer:Answer:

3 (1 2 )(2 )x x x

(2 1)( 2)( 3)x x x

( 3)( 3)( 1)x x x

6 ( 1)( 3)x x x

The order in our answer doesn’t matter, as long as we have all of the factors! The key part to remember is that you need to factorize!

3 24 and 2x x x x 2 23 6 9 and 2 2 12x x x x Answer:

Answer:

( 2)( 2)x x x 6( 1)( 3)( 2)x x x

Adding and Subtracting Fractions

When we add or subtract fractions, we first need COMMON denominators. To do this, we want the least common multiple between the denominators. Then we multiply each numerator and denominator by the factors needed to make this common multiple. Then, we add or subtract the numerators together.

Let me show you an example:

Simplify:

1 2

4 6

Step 1: We need to find the least common multiple between the denominators 6 and 4.

1 3 2 2

4 3 6 2

Step 2: We will multiply the numerator and denominator of each fraction by the factors needed to make 12.

3 4

12 12

Step 3: Multiply the numerators and denominators before adding or subtracting.

3 4 7

12 12 12 Remember, when we add or subtract fractions, we need

the least common multiple. This part is important!

The least common multiple between 6 and 4 is 12.

Step 4: The denominator stays the same and we complete the addition or subtraction to the numerator. We simplify our final answer if possible.

Adding and Subtracting Fractions

Let me show you a second example:

Simplify:

3 7

4 10

Step 1: We need to find the least common multiple between the denominators 4 and 10.

3 5 7 2

4 5 10 2

Step 2: We will multiply the numerator and denominator of each fraction by the factors needed to make 20.

15 14

20 20

Step 3: Multiply the numerators and denominators before adding or subtracting.

15 14 1

20 20 20 Remember, when we add or subtract fractions, we need

the least common multiple. This part is important!

The least common multiple between 4 and 10 is 20.


It is now your turn, simplify the following fractions. Do NOT look at the answers until you have tried them! Also, make sure you show your work and factorize each before simplifying!

Answer: Answer:

Answer: Answer:

67

30

Remember, the key concept is that you need to first find the least common multiple between the denominators!

7 5

5 6

1 3

9 4

1 2 7

4 3 8

6 3

15 10

23

36

1

247

10

Adding and Subtracting Rational ExpressionsWhen we add or subtract rational expressions, the same rule applies. We first need COMMON denominators! To do this, we want the least common multiple between the denominators. Then we multiply each numerator and denominator by the factors needed to make this common multiple. Then, we add or subtract the numerators together. Here is an example:

Simplify:

2 2

2 3

1 2x x x

Step 1: We need to find the least common multiple between the denominators. To do this we must factorize!

2 3

( 1)( 1) ( 1)( 2

( 2) ( 1)

( 2) ( 1))

x x

x x x xx x

Step 2: We will multiply the numerator and denominator of each fraction by the factors that are missing. The order doesn’t matter!

Step 3: Multiply the numerators before adding or subtracting. We can leave the denominators factorized.

5 7

( 1)( 1)( 2)

x

x x x

The least common multiple is (x + 1)(x – 1)(x – 2)


2 4 3 3

( 1)( 1)( 2) ( 1)( 2)( 1)

x x

x x x x x x

Adding and Subtracting Rational ExpressionsHere is a second example:

Simplify:

2 2

5 2

2 3x x x x


5 2

( 2) ( 3

( 3) ( 2)

( 3 )) ( 2)

x x

x xx x x x



3 19

( 2)( 3)

x

x x x

The least common multiple is x(x + 2)(x – 3)


5 15 2 4

( 2)( 3) ( 2)( 3)

x x

x x x x x x

Adding and Subtracting Rational Expressions

Here is a third example:

Simplify:

2 2

3 4

5 4 5x x x x


( 1)

( 1

3 4

( 5) ( 5)( 1))

x x

x x x xx x



3

( 5)( 1)

x

x x x

The least common multiple is x(x - 5)(x + 1)


3 3 4

( 5)( 1) ( 5)( 1)

x x

x x x x x x

Adding and Subtracting Rational Expressions

Here is one more example:

Simplify:

1 3

2 6 3 6x x


3( 2) 2( 3)

3( 2)

1 3

2( 3) 2( 33( 2 ))

x x

x xx x



3 24

6( 3)( 2)

x

x x

The least common multiple is 6(x - 3)(x + 2)


3 6 6 18

6( 3)( 2) 6( 3)( 2)

x x

x x x x

This can be challenging but remember that we MUST multiply the numerators and denominators and then factorize. Always factorize!

Answer: Answer:5 1

2 ( 1)( 2)

x

x x x

Remember, the key concept is that a common factor in both the numerator and denominator can be crossed out and help us simplify the fraction!

2 2

2 3

2 2 2 4x x x x

2 2

2

4 8 3 6

x

x x x x

3 8

12 ( 2)

x

x x

It is now your turn, simplify the following. Do NOT look at the answers until you have tried them! Also, make sure you show your work and factorize each before simplifying!

Answer: Answer:3

( 1)( 1)( 2)

x

x x x


2 2

2 1

1 2x x x

2 2

3

5 6 2 3

x

x x x x

2 4 6

( 1)( 2)( 3)

x x

x x x

Here are some more questions. Simplify the following. Do NOT look at the answers until you have tried them! Also, make sure you show your work and factorize each before simplifying!

Answer: Answer:2 10

4( 1)( 1)

x

x x


3 4

2 2 4 4x x

2 2

1 2

3 12 2 6 4x x x

8 10

6( 2)( 2)( 1)

x

x x x

Here are two more questions. Simplify the following. Do NOT look at the answers until you have tried them! Also, make sure you show your work and factorize each before simplifying!

Simplifying Complex Fractions

Remember a complex fraction will have more complicated numerators and denominators and so each must be simplified first before simplifying the entire fraction. Let me show you an example:

Simplify:

2

3

1

2x

x

Step 1: We must first simplify the numerator and denominator separately. Let me show you.

2

2 3

2

2 3

xx

xx

x x

x x

Step 2: Thus we have the following complex fraction. Remember we have learned how to divide complex fractions. We multiply the reciprocal.

( 2) ( 2) 2

(2 3) (2 3) 2 3

x x x x x

x x x x x

Step 3: We multiply the reciprocal and simplify by factorizing.

2 1 211 x x

We will simplify this part first.

1 221x xx x x

So, we must subtract these fractions using common denominators, which we just learned!

Therefore, our numerator is 2xx

We will simplify this part next.

3 3212 x x 2 3 2 3

1x xx x x


Therefore, our denominator is 2 3xx

Simplifying Complex Fractions

Here is another example:

Simplify:

23

34

3

2

Step 1: We must first simplify the numerator and denominator separately. Let me show you.

73

114

7 4

3 11 Step 2: Thus we have the following complex

fraction. Remember we have learned how to divide complex fractions. We multiply the reciprocal.

7 4 28

3 11 33

Step 3: We multiply the reciprocal and simplify by factorizing.

32 23 1 33

We will simplify this part first.

3 3 9 221 3 3 3


Therefore, our numerator is 73

We will simplify this part next.

3 324 1 42 3 8 32 4

1 4 4 4


Therefore, our denominator is 114

Answer: Answer:1x

Remember, the key concept is that we simplify the numerator and denominator separately and then divide the two fractions.

1

11x

x

x

35

12

2

3

2

5

Here are some questions. Simplify the following. Do NOT look at the answers until you have tried them! Also, make sure you show your work and factorize each before simplifying!

Answer: Answer:(3 2)

( 2)( 3)

x x

x x

Remember, the key concept is that we simplify the numerator and denominator separately and then divide the two fractions.

2

2 12

31

x x

x x

313 4

2 13 2

5

14

Here are two more questions. Simplify the following. Do NOT look at the answers until you have tried them! Also, make sure you show your work and factorize each before simplifying!

Simplifying Rational Expressions – Part II. This PowerPoint contains notes, examples and practice...

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