1. Home
  2. Advanced algebra ii: conceptual
  3. Rational expressions
  4. Multiplying rational expressions

7.2 Multiplying rational expressions

<< Chapter < Page
  Advanced algebra ii: conceptual   Page 1 / 1
Chapter >> Page >
This module covers the multiplication of rational expressions.

Multiplying fractions is easy: you just multiply the tops, and multiply the bottoms. For instance,

6 7 × 7 11 = 6 × 7 7 × 11 = 42 77 size 12{ { {6} over {7} } times { {7} over {"11"} } = { {6 times 7} over {7 times "11"} } = { {"42"} over {"77"} } } {}

Now, you may notice that 42 77 size 12{ { {"42"} over {"77"} } } {} can be simplified, since 7 goes into the top and bottom. 42 77 = 42 ÷ 7 77 ÷ 7 = 6 11 size 12{ { {"42"} over {"77"} } = { {"42" div 7} over {"77" div 7} } = { {6} over {"11"} } } {} . So 42 77 size 12{ { {"42"} over {"77"} } } {} is the correct answer, but 6 11 size 12{ { {6} over {"11"} } } {} is also the correct answer (since they are the same number), and it’s a good bit simpler.

In fact, we could have jumped straight to the simplest answer first, and avoided dealing with all those big numbers, if we had noticed that we have a 7 in the numerator and a 7 in the denominator, and cancelled them before we even multiplied!

Multiplying fractions

This is a great time-saver, and you’re also a lot less likely to make mistakes.

When multiplying fractions...

If the same number appears on the top and the bottom, you can cancel it before you multiply. This works regardless of whether the numbers appear in the same fraction or different fractions .

But it’s critical to remember that this rule only applies when you are multiplying fractions : not when you are adding, subtracting, or dividing.

As you might guess, all this review of basic fractions is useful because, once again, rational expressions work the same way.

Multiplying rational expressions

3x 2 − 21 x − 24 x 2 − 16 ⋅ x 2 − 6x + 8 3x + 3 size 12{ { {3x rSup { size 8{2} } - "21"x - "24"} over {x rSup { size 8{2} } - "16"} } cdot { {x rSup { size 8{2} } - 6x+8} over {3x+3} } } {} The problem
= 3 ( x − 8 ) ( x + 1 ) ( x + 4 ) ( x − 4 ) ⋅ ( x − 2 ) ( x − 4 ) 3 ( x + 1 ) size 12{ {}= { {3 \( x - 8 \) \( x+1 \) } over { \( x+4 \) \( x - 4 \) } } cdot { { \( x - 2 \) \( x - 4 \) } over {3 \( x+1 \) } } } {} Always begin rational expression problems by factoring! Note that for the first element you begin by factoring out the common 3, and then factoring the remaining expression.
Multiplying fractions When multiplying fractions, you can cancel anything on top with anything on the bottom, even across different fractions
= ( x − 8 ) ( x − 2 ) x + 4 size 12{ {}= { { \( x - 8 \) \( x - 2 \) } over {x+4} } } {} Now, just see what you’re left with. Note that you could rewrite the top as x 2 − 10 x + 16 size 12{x rSup { size 8{2} } - "10"x+"16"} {} but it’s generally easier to work with in factored form.
Got questions? Get instant answers now!

Dividing rational expressions

To divide fractions, you flip the bottom one, and then multiply.

1 2 ÷ 1 3 = 1 2 ⋅ 3 = 3 2 size 12{ { {1} over {2} } div { {1} over {3} } = { {1} over {2} } cdot 3= { {3} over {2} } } {}

After the “flipping” stage, all the considerations are exactly the same as multiplying.

Dividing rational expressions

x 2 − 3x 2x 2 − 13 x + 6 x 3 + 4x x 2 − 12 x + 36 size 12{ { { { { size 8{x rSup { size 6{2} } - 3x} } over {2x rSup { size 6{2} } - "13"x+6} } } over { size 12{ { {x rSup { size 6{3} } +4x} over {x rSup { size 6{2} } - "12"x+"36"} } } } } } {} This problem could also be written as: x 2 − 3x 2x 2 − 13 x + 6 ÷ x 3 + 4x x 2 − 12 x + 36 size 12{ { {x rSup { size 8{2} } - 3x} over {2x rSup { size 8{2} } - "13"x+6} } div { {x rSup { size 8{3} } +4x} over {x rSup { size 8{2} } - "12"x+"36"} } } {} . However, the size 12{ div } {} symbol is rarely seen at this level of math. 12 ÷ 4 size 12{"12" div 4} {} is written as 12 4 size 12{ { {"12"} over {4} } } {} .
x 2 − 3x 2x 2 − 13 x + 6 × x 2 − 12 x + 36 x 3 + 4x size 12{ { {x rSup { size 8{2} } - 3x} over {2x rSup { size 8{2} } - "13"x+6} } times { {x rSup { size 8{2} } - "12"x+"36"} over {x rSup { size 8{3} } +4x} } } {} Flip the bottom and multiply. From here, it’s a straight multiplication problem.
= x ( x − 3 ) ( 2x − 1 ) ( x − 6 ) × ( x − 6 ) 2 x ( x 2 + 4 ) size 12{ {}= { {x \( x - 3 \) } over { \( 2x - 1 \) \( x - 6 \) } } times { { \( x - 6 \) rSup { size 8{2} } } over {x \( x rSup { size 8{2} } +4 \) } } } {} Always begin rational expression problems by factoring! Now, cancel a factor of x size 12{x} {} and an x − 6 size 12{ left (x - 6 right )} {} and you get...
= ( x − 3 ) ( x − 6 ) ( 2x − 1 ) ( x 2 + 4 ) size 12{ {}= { { \( x - 3 \) \( x - 6 \) } over { \( 2x - 1 \) \( x rSup { size 8{2} } +4 \) } } } {} That’s as simple as it gets, I’m afraid. But it’s better than what we started with!
Got questions? Get instant answers now!

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Advanced algebra ii: conceptual explanations' conversation and receive update notifications?

Ask