Manipulating Rational Expressions: Simplification and Operations

Are you tired of algebra yet? Come on, it's fun! Before we dive into graphing, let's just do a little bookkeeping with rational expressions. We have to be able to easily simplify, add, subtract, multiply, and divide these, otherwise the more complex algebra that comes later will be impossible. Don't worry, it's not too bad!

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[ProfessorDaveExplains]

Guys, I'm so sorry, the first checking comprehension should have a + 14 rather than - 14 in the second fraction, that makes it work out nicely. And the second problem is also looking strange and you may have to disregard it. I can't believe I let these errors slide, rest assured the other math comprehensions are not like this!

[webdev]

it takes me days and lots of research... then I see on the comment line and the question was wrong. ain't regret it tho. it makes me learn more.

[tenoutofzen]

SOLUTION TO PROBLEM 2:

I hope you find this comment, because I'm here to say that if you weren't able to solve the second problem is cause it's written wrong. "3 / x+5" is NOT the correct answer to the exercise. And you should not feel like you're falling behind just cause you can't seem to get to the same answer written on the screen. People make mistakes and that's okay, it happens. As Prof. Dave wrote "-14" instead of "+14" in the first exercise, he also wrote something wrong in the second exercise, or simply solved it wrong. I stayed at this exercise trying to get to the same answer from yesterday. Took me so long to get to the incorrect answer until I realized it's just something there that's written wrong. Just move on, there's nothing wrong with you. Keep studying, cause if you've got to the point of reading explenations to one problem of one random math video on Youtube it means that you're dedicated. And you should stay that way. You're definitely on the right track.

[thermalrain_yt]

Hooray for algebra. Started this 3 weeks ago during my machine runtime at work and made it this far. Half a notebook used up. You're a great teacher.

[arsaoramba]

I passed my test because of him. glad i subscribed

[NickHope]

COMPREHENSION #1 SOLUTION
Bearing in mind that '- 14' should have been '+ 14'...
((x^2 + 3x - 4)/(x^2 + 4x - 21)) ÷ ((x^2 + x - 2)/(x^2 + 9x + 14))
= ((x + 4)(x - 1))/(x + 7)(x - 3)) ÷ ((x + 2)(x - 1)/(x + 2)(x + 7)) - factorization.
= ((x + 4)(x - 1))/(x + 7)(x - 3)) * ((x + 2)(x + 7)/(x + 2)(x - 1)) - changing from ÷ to * and flipping the fraction on the right.
= (x + 4) (x - 1) (x + 2) (x + 7) / (x + 7) (x - 3) (x + 2) (x - 1) - shown more simply as 1 fraction with terms combined (multiply numerators together and denominators together).
= (x + 4) -(x - 1)- -(x + 2)- -(x + 7)- / -(x + 7)- (x - 3) -(x + 2)- -(x - 1)- - cancelling matching factors on the top and bottom.
= (x + 4) / (x - 3)

[Iomhar]

Tsk-tsk-tsk!
Well, I figured out that there should have been a +14 instead a ‒14 in the first expression because there is no way that a+b=9 and a×b=‒14. However, I thought I might be wrong and tried to work it the other way round, from the solution to the original. It didn’t work, so I was even surer about the mistake.
The second one was trickier. I tried to work it from solution to the original expression and there was no way for it to work.
I am sorry that I didn’t read the comments before starting doing the maths here.

[bathtubanarchy]

At 5:27, I feel like I'm missing the point of getting the lowest common denominator? Why would you not cancel out terms from each expression first and then work from there? I.e (x-5)(x+1) / (x+9)(x+1), cancel out (x+1), leaving (x-5)/(x-9), and do the same to (x+2) in the other term, so you have

[(x-5)/(x-9)] - [(x-3)/(x+1)] and work with that?

[peaceofmind_ytc]

Thanks for your videos, sir. I learned a lot from them. 😊😊

[sattaralramahy]

5:47
we can just multiply the Numerator an the Denominator by x
so it will give us a direct answer to the simplifying

[tensaijuusan]

I just spent 30 minutes tearing my diminishing hair out - I had been telling my Math teacher wife just this morning that this guy must be a genius or polymath of some kind. She kind of sneered at me!

[Huwskie]

I didn’t check the comments and was trying to solve for when it was -14 all three ways. Kinda funny

[noapsi]

wait why cant we just cancel out 1+ square root 2 at 8:06?

[abhijitchakraborty]

Thank you so much sir for the lovely video

[ViniciusRaphael]

Solution 2:
- (3x) / (x + 5) - (6) / (x - 2)
- (x - 2) * ( 3x) / (x - 2) * ( x + 5) - (6) * (x + 5) / (x - 2) * (x + 5)
- (3x^2 - 6x) - (6x + 30) / (x - 2) * (x + 5)
- (3x^2 - 6x - 6x - 30) / (x - 2) * (x + 5)
Result: (3x^2 - 12x - 30) / (x - 2) * (x + 5)
Same as: 3(x^2 - 4x - 10) / (x - 2) * (x + 5)

[ahnafalnafis]

Could someone solve problem #2? I left with 3(x^2 - 4x - 10) over (x + 5) (x - 2) :(

[amarynthia]

Professor, is the second problem wrong too? I really can't solve it. Can you give the explanation on this one.

[paradox]

Why don't we like radicals in the denominators of fractions?

[Attalla-tb]

4:44 Why don't we FOIL the denominator?

[HellooDae]

I appreciate the work, but those mistakes really messed me up. 1 or 2 hours down the drain trying to figure out what did I get wrong.