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

Simplifying Rational Expressions – Part I

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, when I have a number I need to find all of its prime factors. It can be quite easy but let me show you an example.

Given then number below, what are its prime factors?

20

We know that 4 multiplied by 5 equals 20, so 4 and 5 are factors of 20 but 4 is NOT a prime number. So, we must find the prime factors of 4 also! This is easy, the prime factors of 4 are 2 multiplied by 2. So, we have:

20 2 2 5

Therefore, are the prime factors of 20 and we are done!2 2 5

So, we are looking for prime numbers which, when I multiply them together, I get my answer.

It is now your turn, find the prime factors of the following numbers. Do NOT look at the answers until you have tried them!

45 32

30 24

Answer: Answer:

Answer: Answer:

3 3 5

3 2 5

2 2 2 2 2

2 2 2 3

The order in our answer doesn’t matter, as long as we have all of the numbers!

28 70

Answer: Answer:

2 2 7 2 5 7

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 prime 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 are the factors of the expression below:

3 23 6x xSo, we are looking for algebraic factors which, when I multiply them together, I get my answer.

Guess what, we are factorizing again! We have several methods of factorizing that we can use and MUST understand. The first method is to look for any factors which are common in each term.

Notice, we have in common with both terms and so must factorize it using that.

23x

3 2 23 6 3 ( 2)x x x x And this is our final answer!

Here is some practice, find the factors of the following expressions. Try them first before you look at the answer and don’t move ahead until you are confident you know what to do!

24 8x x 2 4x

2 36 9x x 25 10x

Answer: Answer:

Answer:Answer:

4 (1 2 )x x

23 (2 3 )x x

( 2)( 2)x x

25( 2)x

The order in our answer doesn’t matter, as long as we have all of the factors!

2 5 4x x 22 5 3x x Answer:

Answer:

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

Here is some more practice, find the factors of the following expressions. Try them first before you look at the answer and don’t move ahead until you are confident you know what to do!

22 8x 2 9x

26 7 2x x 3 22 2 4x x x

Answer: Answer:

Answer:Answer:

22(1 4 )x

(3 2)(2 1)x x

( 3)( 3)x x

2 ( 1)( 2)x x x

Remember, the order in our answer doesn’t matter, as long as we have all of the factors! It is important that you can factorize WELL, so practice!!!

3 22 10 12x x x 33 3x xAnswer:

Answer:

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

So, we use this concept of factors when we simplify fractions!

I am sure that most, probably all, of you are very comfortable with simplifying fractions but I will explain the concept and ask you to practice it because it is important to understand when we begin simplifying rational expressions.

The first step we do when simplifying fractions is factorize the values in both the numerator and denominator. This means we factorize both the top and bottom parts of the fraction.

After we factorize it, if we find a common factor in both the top and bottom parts, then we can cancel it out. A common factor will divide to one and this can cancel!

Most of you can probably do this in your head with no work, but let me show you an example on the next slide. It is important that you understand the concept, so do NOT go on until it makes sense!

Here is an example with the steps included. See if you can do each step before looking at the answer. Simplify the fraction below:

20

12Step 1: We first factorize both the numerator and denominator. We do not need prime factors (although this is okay) but we are looking for common factors.

20 4 5

12 4 3

Step 2: Now that we have factorized each part, we see a common factor of 4. We can cross this out as 4 divided by 4 is 1.

20 4 5

12 4 3

Step 3: We now look at our new fraction and see if we can find any additional common factors in both parts. We cannot and so we have our simplified fraction!

5 21

3 3

Either answer is fine.

Easy! I am sure most of you can do this without writing anything down but practice this concept of factorizing and removing common factors as it is important to understand.

Final Answer:

Here is a second example. Simplify the fraction below:

25

30Step 1: We first factorize both the numerator and denominator. We do not need prime factors (although this is okay) but we are looking for common factors.

25 5 5

30 5 6


25 5 5

30 5 6


5

6Easy! I am sure most of you can do this without writing anything down but practice this concept of factorizing and removing common factors as it is important to understand.

Final Answer:

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:

3

2

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!

27

18

22

44

12

28

24

30

1

2

3

74

5

Now that we have discussed how we use factors to simplify a fraction, we can apply this to simplifying rational expressions.

But, what is a rational expression?

Basically, a rational expression is a fraction which contains variables in either, or both, the numerator or denominator.

Here are several examples:

2 2 3

2 2

2 5 1 3 or or

30 2 2 2

x x x

x x x x x

These are all examples of rational expressions. There are MANY more examples of rational expressions.

Now, let us look at how factorizing can be used to simplify a rational expression.

Here is an example with the steps included. See if you can do each step before looking at the answer. Simplify the fraction below:

22 6

12 10

x

x

Step 1: We first factorize both the numerator and denominator. We CANNOT cross anything out until we factorize. Remember, we MUST factorize!

2 22 6 2( 3)

12 10 2(6 5)

x x

x x



2 3

6 5

x

x

We cannot cross anything else out because we CANNOT factorize either part. So, only cross out common factors.

Final Answer:

2 22 6 2( 3)

12 10 2(6 5)

x x

x x

Here is a second example with the steps included. See if you can do each step before looking at the answer. Simplify the fraction below:

2

2

1

2

x

x x


2

2

1 ( 1)( 1)

2 ( 2)( 1)

x x x

x x x x

Step 2: Now that we have factorized each part, we see a common factor of (x + 1). We can cross this out as (x + 1) divided by (x + 1) is 1.


1

2

x

x


Final Answer:

2

2

1 ( 1)( 1)

2 ( 2)( 1)

x x x

x x x x

Here is one more example with the steps included. See if you can do each step before looking at the answer. Simplify the fraction below:

3

3 2

3 12

3 9 6

x x

x x x


3

3 2

3 12 3 ( 2)( 2)

3 9 6 3 ( 2)( 1)

x x x x x

x x x x x x

Step 2: Now that we have factorized each part, we see several common factors. We can cross these all out.


2

1

x

x


Final Answer:

3

3 2

3 12 3 ( 2)( 2)

3 9 6 3 ( 2)( 1)

x x x x x

x x x x x x

This can be challenging but remember that we MUST factorize first. Always factorize!

Answer: Answer:3

2

x

x


2

2

2 6

2 4

x x

x x

2

2

3 4

6 8

x x

x x

1

2

x

x



Answer: Answer:2 1

1

x

x


2

2

2 1

1

x x

x

2

2

2 10 12

2 8 6

x x

x x

2

1

x

x



Answer: Answer:3

3

x

x


3 2

3 2

5 6

6

x x x

x x x

2

3

2 2 4

6 6

x x

x x

2

3 ( 1)

x

x x


Multiplying Fractions

When we multiply fractions, we multiply the numerators together and multiply the denominators together, or we multiply the top numbers together and bottom numbers together.

The key part to remember is that we actually only need to multiply the FACTORS together and then, if we find a common factor, simply it. Let me show you an example:

Simplify:

4 20

15 12

Step 1: We will multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you.

4 20

15 12

Step 2: We will factorize our numbers, looking for common factors.

4 20 4 4 5

15 12 3 5 3 4

Step 3: Any common factors on the top and bottom can now be crossed out and the rest of the fraction simplified.

4 20 4 4 5 4 4

15 12 3 5 3 4 3 3 9

Therefore, when we multiply fractions, before we multiply, we can factorize the numbers and cross out any common factors.

Here is one more example:

Simplify:

9 15

10 12

Step 1: We will multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you.

9 15

10 12


9 15 3 3 3 5

10 12 2 5 3 4


9 15 3 3 3 5 3 3 9

10 12 2 5 3 4 2 4 8

Therefore, when we multiply fractions, before we multiply, we can factorize the numbers and cross out any common factors.


Answer: Answer:

Answer: Answer:

2

3


12 20

15 24

8 27

9 28

6 15 9

10 12 8

12 30

15 16

6

7

27

323

2

This same method can be applied when we multiply rational expressions. We multiply the numerators and denominators, then FACTORIZE. Here is an example:

Simplify:

2 2

2 2

2 6 1

3 2 3 3

x x x

x x x x

Step 1: We will multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you. Notice that I needed parentheses!

2 2

2 2

(2 6 )( 1)

( 3 2)(3 3 )

x x x

x x x x

Step 2: We will factorize our expressions, looking for common factors. Notice again the parentheses!

2 ( 3)( 1)( 1)

( 1)( 2)3 ( 1)

x x x x

x x x x


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

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

x x x x x

x x x x x

Therefore, when we multiply rational expressions, we factorize and cross out any common factors.

Here is a second example:

Simplify:

2 2

2 2

9 3 2

6 2 3

x x x

x x x x

Step 1: We will multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you. Notice that I needed parentheses!

2 2

2 2

( 9)( 3 2)

( 6)( 2 3)

x x x

x x x x


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

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

x x x x

x x x x


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

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

x x x x x

x x x x x



Simplify:

2 2

2

6 18 2 8

6 9 36

x x x x

x x x

Step 1: We will multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you. Notice that I needed parentheses!2 2

2

(6 18 )( 2 8)

( 6)(9 36)

x x x x

x x x

Step 2: We will factorize our expressions,

looking for common factors. Notice again the parentheses!

3 2 ( 3)( 4)( 2)

( 4)( 2)3 3( 3)

x x x x

x x x

Step 3: Any common factors on the top and bottom can now be crossed out and the rest of the fraction simplified. Lastly, we put the number in front of the rest of the expressions.

Again, when we multiply rational expressions, we factorize and cross out any common factors.

2

2

6 ( 3)( 2 8)

( 6)9( 4)

x x x x

x x x

3 2 ( 3)( 4)( 2) 2 ( 3)

( 4)( 2)3 3( 3) 3( 3)

x x x x 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:2

2( 1)

x

x


2

2

4 8 5 6

2 3 8 16

x x x

x x x

2 2

2 2

3 1

4 2 3

x x x

x x x x

1

4

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:2

3 2

x

x


2 2

2 2

2 3 1 4

3 5 2 2 3 2

x x x

x x x x

2 2

2 2

2 2 9

4 8 12 2

x x

x x x x

3

2( 2)

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 3

3 2

x

x


2 2

2 2

2 6 3 4 1

3 5 2 3 2

x x x x

x x x x

3 2 2

3 2

2 2 7 3

2 3 2

x x x x x

x x x x

3

1

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!

Dividing Fractions

When we divide two fractions, we MULTIPLY the first fraction with the reciprocal of the second fraction.

After we take the reciprocal, the key part to remember is that we actually only need to multiply the FACTORS together and then, if we find a common factor, simply it. Let me show you an example:

Simplify:

4 20

9 12

Step 1: We will first multiply the first fraction with the reciprocal of the second and then multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you.

4 12

9 20


4 12 4 4 3

9 20 3 3 5 4


4 12 4 4 3 4 4

9 20 3 3 5 4 3 5 15

Therefore, when we divide fractions, we multiply the first fraction with the reciprocal of the second fraction.

4 12

9 20


Simplify:

610

415

This is called a COMPLEX FRACTION and is basically the top fraction divided by the bottom fraction.

Step 1: We will first multiply the top fraction with the reciprocal of the bottom and then multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you.

6 15

10 4


6 15 3 2 3 5

10 4 2 5 2 2


6 15 3 2 3 5 3 3 9

10 4 2 5 2 2 2 2 4

Therefore, when we divide fractions, we multiply the top fraction with the reciprocal of the bottom fraction.

6 15

10 4

6 4

10 15


Answer: Answer:

Answer: Answer:

4

3


16 24

5 10

12 20

9 27

1012

159

149

216

9

5

1

24

9

Again, this same method can be applied when we divide rational expressions. We multiply the first with the reciprocal of the second, then FACTORIZE. Here is an example:

Simplify:

2 2

2 2

12 6 8

2 1 2

x x x x

x x x x

Step 1: We will first multiply the first fraction with the reciprocal of the second and then multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you. Notice that I needed parentheses!

2 2

2 2

( 12)( 2)

( 2 1)( 6 8)

x x x x

x x x x


( 4)( 3)( 1)( 2)

( 1)( 1)( 4)( 2)

x x x x

x x x x


( 4)( 3)( 1)( 2) 3

( 1)( 1)( 4)( 2) 1

x x x x x

x x x x x


2 2

2 2

12 2

2 1 6 8

x x x x

x x x x


Simplify: 2

2

2

2

2 64 4

6 62

x xx x

x xx x

This is called a COMPLEX FRACTION and is basically the top fraction divided by the bottom fraction.

Step 1: We will first multiply the top fraction with the reciprocal of the bottom and then multiply the numerator and denominator together and as one fraction. We don’t actually need to do the math, just write it out. Let me show you. Notice that I needed parentheses!

2 2

2 2

(2 6 )( 2)

( 4 4)(6 6 )

x x x x

x x x x


2 ( 3)( 2)( 1)

( 2)( 2)3 2 ( 1)

x x x x

x x x x


2 ( 3)( 2)( 1) 3

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

x x x x x

x x x x x


2 2

2 2

2 6 2

4 4 6 6

x x x x

x x x x

Answer: Answer:3 2

3

x

x


2 2

2 2

6 7 2 5 6

2 3 1 3 2

x x x x

x x x x

2 2

2 2

6 12 9 9

2 7 6 2 3

x x x

x x x x

2

3( 1)

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:1

4

x

x


2

2

2

2

13 2

162 8

xx xxx x

2

2

2

2

4 9 22 5 24 11 32 1

x xx xx xx x

1

3

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

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