Simplifying Rational Expressions… How? (NancyPi)

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Published on July 2, 2018 by

For how to FIND the DOMAIN of a function, jump to https://youtu.be/GQGFMUfr10M
For more factoring examples in my video on how to solve quadratic equations by FACTORING, jump to: https://youtu.be/Z5MnP9da4EM

There are three steps to simplify algebraic rational expressions:
1) Factor the numerator and denominator as much as possible.
2) Cancel factors that are the same on top and bottom.
3) Write a restriction ("condition") for any x values that used to make the denominator zero, before you cancelled factors. For instance: "x cannot equal -1".

1) FACTOR as much as you can: Factor the top and bottom as much as possible. In both the numerator and denominator, always check first if there's a number or a variable (x) that is in all the terms that you can pull out front. If you have something that looks like "x^2 minus a perfect square number", like x^2 - 9, you can factor using the difference of squares formula, into (x + 3)(x - 3). If you have a difference of cubes (x^3 - 8) or sum of cubes (x^3 + 27), you can factor using the sum/difference of cubes formulas. If you have a quadratic expression on top or bottom that has three terms ("trinomial"), you can factor it using trial and error, or the "magic x" trick for factoring. To learn how to factor quadratics, check out my video on factoring.

2) CANCEL: Cross out any factors that are the same on top and bottom, and write what's left behind. When you've cancelled everything you can, you now have your simplified rational expression, or expression "reduced to lowest terms" or in "simplest form".

3) CONDITION: Is that all? To be technically correct, you should also write a note next to the simplified rational expression that states the restriction. You can check with your teacher to see if you’re required to write it for your class. But either way, the restriction is there, and the final answer isn’t really complete without this “condition”. All you need to do is: look at any factor(s) you cancelled in the denominator (ex: x-7), see what x value used to make that factor equal to zero (7), and next to your answer write “x not equal to” that number (x not equal to 7).

We put the restriction, or condition, there at the end, because if we didn’t, the restricted x value would appear to be a valid x value for the expression, when it’s not. When we simplified the expression by cancelling the bottom factor, we got rid of the undefined, division-by-zero problem for that x-value, but in reality that x-value is still not allowed. So to be totally correct, write the note that states this restriction next to your answer.

For how to find the HORIZONTAL ASYMPTOTE of a rational function, jump to https://youtu.be/qJrrZQgSkO8

For more math help and videos, check out: http://nancypi.com

Edited by Miriam Nielsen of zentouro: https://www.youtube.com/zentouro

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• first to comment….. Great video…..This is very helping…..

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• Hi Nancy, thanks for the great video. I am having trouble with factoring the final example when you get (5x-6)(3x+2). Do you know wher I am going wrong or where to look for assistance

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• I do not understand how you factorize the terms ? Just guessing ? Why 5x-6 ? Just quessing ? Why not 5x-3 ?

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You got me
Through trig and calculus! Thank you!!

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• Thanks for teaching me how to simplify a rational expression.
Angel Nancy,
Why is 1/0 undefined? Please explain me.
2nd example contains a hole at X=1 and also there is another restriction at X=-6.
Nancy,Isn’t it?
GOOD NIGHT!

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• Hi @NancyPi, can you make video on definition of polynomial? i.e. what makes a algebraic express a polynomial.

• I had no idea your original conditions still remain after simplification, I thought the purpose was to adjust the expression so that there would no longer be a condition!!! Thank you Nancy, i’m glad you’re back.
* and I’ve never seen this hilarious quirky side to you! 3:54 — love it.

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• A couple of questions about the last example: Where does the -10 come from? Doesn’t -6+2=-4? How do you get -8 from that? Please forgive my ignorance—I REALLY struggle with math.

• She factored out -2 from 20, giving negative -10

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Thanks for all the help so far NacyPI

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Not like she wants u wake up bruv

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Enjoyed this first video of yours regarding “rational expressions”. I hate algebra & really don’t see the need for it on a everyday basis but
it’s a graduation requirement, unfortunately. Thanks for your simplified & direct problem solving approach! I will watch your other videos!

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• What if you have a negative rational function that is on the numerator, not too sure, plz help

• with the last example, how did you get the 6 and 2 in (5x-6)(3x+2)?? Don’t the two numbers have to equal 12 when multiplied and 8 when subtracted?…..so 6-2=4?….I’m confused…and it wouldn’t be 4 and 3 bc 4-3=1 and 12-1=11….

• Exactly!!!

• ok ALSO, on example 3, you “getting lucky” means you put (5x-6)(3x+2) in the correct order? I initially wrote it as
(5x+2) and (3x-6) and was wrong….

• ALSO on example 3 LOL sorry, why did you choose 5 & 2 and not 10 & 1? what’s the process to figuring that out?

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• Forgive me, but on the last one you can also do 15x and x which gives you 15x squared as well, HOW DO YOU KNOW WHICH ONE TO CHOOSE?!

• -6 + 2 = -4 and -6 + 1 = – 5 for Q3) both multiply to get C term but do not add to B term. Tried to FOIL it as well but can’t

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• 12:30 /6 and -2 don’t add up to -8? was that an error, and they were both supposed to be negative?

• Yes I’m confused on this

• Imo that works only if there is no coefficient before x. In the case you are talking about, we have 5x and 3x, which means that you have to check the answer by doing (5x*2)+(-6*3x)=10x-18x=-8x

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(edit: totally kidding)

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Super

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• KOT3Q That’s easy to answer. You see when you get older or more mature, it’s important to keep the brain 🧠 healthy. So good health is good diet, exercise, maths and lots of making out. So in a nutshell, keep on living like a 20 year old till you drop.

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• Hello Nancy!
how can we use Magic X for (10x^2 -7x -6)? while we can’t multiply 1 & -6 for -60 that becomes from a*c [in ax^2 + bx + c] and add -6 & 1 for -7 it becomes from b [in ax^2 + bx + c]
I don’t undrestand :'(

• Cant help wonder ? Depending on your answer ,if you have one,or respond, will become relative to me asking you another.
What were your thoughts , if any, regarding the non cloning theorem for teleportation, when you first observed that the answer to the theorem was claimed to be hidden from human knowledge ?

• When she said “you don’t have to live like that”😂

• what a beautiful teacher,,

• Sample #3…. what is the rule for inserting the -6 with the (5x ) instead of the +2 ? Which would have been (5x+2)(3x-6).

• Because developing (5x+2)(3x-6) the result is 15x^2-24x-12 which os different from 15x^2-8x-12 (original equation)

• @Leonardo Miguel
Perfect, got it…. thanks

• ik this is really dumb but can anyone further explain the factoring portion of the third pproblem
15x^2-8x-12 and how it factored to what Nancy got

or at least link a vid to help my dumb self?

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• Nice video. Hope to see more lectures.

• Nancy, you had me at ‘You don’t have to live like that.’ <3

ETA: It's not true that x<>7 is required in Example 1 because there is not actually a singularity at x = 7. You know this (although a student at this level might not). Similarly with Examples 2 and 3. I think it’s true in general when you factor out a zero/zero binomial, as long as the surviving denominator doesn’t have a zero in the same place.

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• the third one has an oopsy in it … (-6+1 is not = to -7 ) its =–5 and ( -6+2 is -4 not -8 ) but any ways thanks for helping us ❤

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• simple to simplify!!

• What happend to the two?

• I am sorry but in the last exercise you throw me out on the top simplification and the compositive quadrastic ….
I know you make this exercises with your eyes closed, but please why there is a ÷6 and -2 ; if you added is -4 not ÷8

• you did great for the explanation

• Nancy can you tell us few words on your background? Very good videos. Tks

• 13:07
I do not undrstand how she goes from 5x = 6 to x= 6/5?

• She divided the x term and the 6 by 5 to allow a restriction to occur in simplest terms

• @LopsidedEdits Thanks 🤙

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• 10:40 ON THE TOP U CAN FACTOR 3….

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• Isn’t the first problem x not equal to -7?

• Can you post a video on multiplying, dividing, adding, and subtracting rational expressions!

• Hi NancyPi, thanks for your videos. I need a sol. Of an eq.
Could you find the value of X please in this eq. : sqrt(5-4x^2+8x)+sqrt(8x-4x^2)=9-8x+4x^2 .

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• #nancypi you are the best, I understand every single video you upload. I have an A in math class because of you thank you! My teacher always complicates thing, and is not very clear.

• Nanci Pi, MIT graduate, you’re smart and make Math beautiful just like you! You make math fun and doable. However, my son’s teacher will throw in exotic problems such as this: (2x-5)(x+3)/15-3x less or = 0. Where he has to graph the solution set and write it using interval notation. We have asked several people with our circle who are Math majors who get stumped because the haven’t had to do such problems in a long time.

Teachers like the one my son has give Math a sour taste for kids who are already struggling. What is most frustrating to my son is that the teachers seems to work the easier problem in class on the board, but leave the more difficult ones for homework. Isn’t the goal to win kids to Math? Thank you in advance for helping to sweeten things up a bit for my son and so many others. 🙏 😃

Signed,
A concerned, yet hopeful mom who wants her kids to love Math like you do! 🥰🤗❤️

• I usually watch all my youtube videos on x2 speed so when i put it on normal after watching one already sped up, I realized how seductive her voice sounds lol. Random. helpful video though!

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• Nancy are you writing backwards or is there something else involved? Can you answer this question once and for all? lol also thanks great vid!

• 9:55 wholesome moment

• We do not really “cancel the factor”. Rather we multiply by 1/(x-1) in the numerator and in the denominator. By doing that nothing changes since it comes back to multiplying by 1, if the denominator is different from 0.

• Exercise 3, x cannot equal -1/2 neither.

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• Can you please make a video on solving rational equations 😫🙏🏼

• This woman is smarter than me at my best.

I’m not simping, no cap.

• thank u .. for the video

• Me to Nancy: Wanna hangout tonight?
Nancy: 03:57

• YES!! That was good! I love it. Take a like from me.

• Nancy, I’m with Grover on the last problem. -6 +2= -4 not -8?

• thanks you Nancy!!!

• You are a great mathematician! You so fast at times it freaks me out! I’m older so I am just looking to re-fresh myself. You are an excellent teacher!!

• Hy

• your explanations are great !! I get the understanding ..thank you

• Love ur video

• Your explanation is great. The numerator is easy to factor. This how i will factor the numerator. Is (X+7)and (x_7) which is _49 For my denorminator. I will use. 2 and negative _ 7 and so forth to solve the denominator. My own short distance to solve that expression. Thank you my dear Nancy. Can you make video to work on. trynomo please. Your explanation is great trust me.

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• Ahh thank u! This helped A LOT 🙂

• hey nancy i failed 5 times in math i dont know why i watched every single video of your classes any ways now im studying arts love your classes still

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• Cancel a can?

• That was very useful.

• I don’t get what you do with -8 please

• Can you do a video on adding and multiplying rational expressions?

• Gracias por la información de la reducción buen método

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• and pretty

• 0:22 oooo I think I heard some Nancy humor in there

• Nancy in equation no. 3 how that -6+2=-8 ?

• She has such a nice voice

• Is having ure hears rightside a symbol for an obsessed clan/cult? Ouh goddamn u got busted by me. I am gonna send u an swat.

• I’m gonna cry I’m so lost

• Can’t do calculus until you learn these concepts thoroughly 🙂 Thanks & thumbs up 👍

• Slow down with the instructions because am using a computer too just letting you know smart lady love ur video with passion

• Great video. How do you record the transparent transparent board. Can you describe your video setup. I’m an instructor myself and I would like to implement a similar method for my students in CIS

• I’m not even in school anymore I just love watching her videos for fun 😌

• 0:58 yeeeeeesss, please show me for the complicated cases of functional equations. it seem like rote learning the existing methods to me. I cant quite get my head around it.

• “If you’re one of the wise ones you’ll stick around.”

Me that’s definitely not wise but definitely still needs to stick around

• I have a question please I need help with it
if y=2×4 and h=4x+6 fine a and y