Simplifying Rational Expressions... How? (NancyPi)

Description: MIT grad shows how to simplify a rational expression. To skip ahead: 1) for how to simplify an expression with DIFFERENCE of SQUARES factoring in it (x^2 - 49), skip to time 1:01. 2) for how to write the CONDITION or RESTRICTION, skip to time 3:52. 3) for how to simplify an expression with QUADRATIC(s) in it (like the x^2 - 3x + 2 trinomial), skip to time 6:12. 4) for an example with LARGER COEFFICIENTS and a NEGATIVE x^2 term (like -20x^2 + 14x + 12), skip to time 9:57. Nancy formerly of mathbff explains the steps:

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.

[sam]

Absolutely love your explanations. I learned all this 65 years ago; you're a great refresher and you fill in a lot I never heard.

[paulpritchard]

I am self studying mathematics here and your videos are a goto resource for me as your ability to explain complex concepts simply, and your great conversational teaching style, really gets the idea across.

Thanks for all the help so far NacyPI

[Playgrl]

You need to be a math teacher. You have a natural gift. You not only show how you do the math, but you explain it very well so that you understand how to rationalize the steps though the problem.
You got me
Through trig and calculus! Thank you!!

[TheZooman]

I am in my third year of Electrical Engineering at ENMU, and I am watching this. Nancy makes Math fun. I could watch math for hours... I often refer to the video on Logs... we use Logs a lot.

[Onihikage]

It's good to hear your voice again, Nancy! I love what you've done in the description, it's like a typical math course session, and definitely more search engine friendly for struggling students. Someday your channel could be all a body needs to learn advanced math!

[denhayy]

The reason I got into a selective school was 85% from these maths tutorials. Thank you so much!

[elioraa.]

Thank you for showing how you did the factoring! My teachers just skip over those details, and I have trouble factoring, and this helps me remember how to do it with different types of equations. Thanks!! These math videos are helping so much!

[skullcandy]

I feel like in a relationship, you don’t want to argue with Nancy. She’ll always win.😂 5:49

[sheluvdee]

Thx nancy ur vids really help me out, im studying that in school and i've gotten 85's and 95's on test! Thank you so muchh nancy ur the best

[heartforsale]

Nancy you're such an amazing teacher, I've never ever found explanations of maths concepts so engaging and accessible. Thank you so much!

[drscott]

Your videos have really become much more funny over the year. I like your humor

[EBestLetterE]

it’s always nice to listen to a beautiful math teacher

[diojan]

I don't know why there's about 2% who disliked this tutorial. I find this tutorial very very good. Thanks NancyPi for making videos like this.

[frankthompson]

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!!

[chelseaaao]

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

[hermitcard]

Always a relief to be able to pause, rewind and watch many times a video; instead of a real time class 🙄

[Nl]

I mean absolutely no disrespect whatsoever, but I find myself watching your videos and learning math simply because I find you beautiful, enjoyable, intelligent, and engaging. Keep up the great work. You are an inspiration!!

[JReed]

I'm glad you are still making videos, the new girl on MathBFF is alright but she is no NancyPi.

[dannycagley]

You helped me tremendously through school. Just graduated this spring with a 3.97 Summa Cum Laude! Thanks for your great help! I recommend you to everyone I know that needs math help. You are awesome!

[thejerrycheng]

Thanks for coming back Nancy! I found your calc lectures really helpful