06 - Simplifying Rational Expressions in Algebra, Part 1

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In this lesson, you will learn what a rational expression is in algebra and how to simplify rational expressions. A rational expression is any expression that can be written as a ratio of two polynomials.
In a similar way, a rational number can be written as a fraction with numerator and denominator, a rational expression can be written with a numerator that is a polynomial and a denominator that is also a polynomial.
Here we focus on how to simplify these types of expressions, which usually involves factoring the numerator and denominator and canceling common terms.

[@firetalongames91]

I'm in algebra 2 and my teacher doesn't explain any of this. He shows us him doing a problem and then we're given an assignment. Watching these videos are such a big help.

[@mathfullyexplained]

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[@chinmaibommana9788]

Wait, this is in algebra 2?

[@firetalongames91]

@@chinmaibommana9788 for me it was. Idk how it will be for you. If it is, good luck

[@chinmaibommana9788]

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[@tomjoad6993]

12:15 I just now realized what factoring really is and what it can be used for. Factoring is rewriting a number or expression so that it is the product of multiplication. And one reason for doing this is so you can cancel out terms in a fraction in order to quickly simplify it. You cannot cancel common terms between the numerator and denominator if there is addition or subtraction involved between any of the terms, so factoring it makes everything the product of multiplication instead of addition / subtraction, which then allows you to cancel terms. I really love when things start coming together and everything becomes clearer.

[@tomjoad6993]

When I first started learning Algebra, which was only about a week ago. This is the first question I asked when something didn't make sense to me: "I was watching a video that says 9x + 3 / 12x + 4 can be simplified to 3/4. But can't just 9x / 12x be simplified to 3/4? Can't 9x + 3 / 12x + 3 also be simplified to 3/4? How can they all be the same thing when they are clearly not the same thing? I understand how 9x / 12x and 9x + 3 / 12x + 4 are the same thing. But what about 9x + 3 / 12x + 3? Isn't that also 3/4? Since don't the + 3's on each one cancel each other out? " Do you see at the end where I said, "Don't the 3's cancel out?" And they don't because they are being added to the other terms not multiplied.

[@tomjoad6993]

Something else I just realized, too. When you flip terms that are being added or subtracted by multiplying by -1 you are simply factoring it! I was wondering where you get the -1 from. But all you are doing is finding a way to make it the product of multiplication. It's like taking a single number 12 and rewriting it as 3(4). Where did you get the 3 and 4 from? You simply made 12 the product of multiplication by using its factors! THAT'S WHY THEY CALL IT FACTORING, JASON! I HOPE YOU LEARNED SOMETHING NEW TODAY. :)

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[@georgeh1352]

Neat trick using ( -1), wish I had been taught that at school/college. Pot luck on the quality of Teacher / Lecturer you get, thanks for your clear and top notch explanations.

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My son's teacher just keeps piling on the homework, but many kids dont understand fully, including himself. Makes no sense. My son said you're better at teacher than his teacher. Thankyou

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[@blaneferris9952]

It's helps so much that you rewrite the problem before cancelling, other videos look so complicated when they do it all on top of the original problem.

[@PalenVencatachellum-xo3fc]

I thank you very much for your explanation which is very simple to assimilate by anybody.

[@exotickhanaarif4333]

Wow man that -1 is such an important tool it helped me understand what Factoring really is

[@marianusStaff]

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[@MathAndScience]

You are very welcome! View all my Lessons and Courses! Visit: www.MathTutorDVD.com

[@dusannovakovic5257]

Great lecture. If only there were more professors like you, especially for math and sciences. There is an interesting point I never thought of until now. Maybe you could've used the example with s^2-t^2/(s-t)^2 to show explicitly that (s-t)^2=(t-s)^2 which is pretty convenient to keep in mind when solving similar equations because such manipulation makes the process faster.

[@mathfullyexplained]

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It was useful session :)! I like the sessions from Vidya Guru channel as well. Their important questions were proven very helpful in my SSC exam.

[@mathfullyexplained]

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neat writing and clear explanation. i like it

[@thedoctordowho2022]

Thank very much for you time to do this lessons.thank you!!!

[@mathfullyexplained]

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[@markfinn825]

Some numbers with decimal points can not be written as fractions . But if the circumference of perfect circles are divided by their exact diameter is the number always the same ( no matter what the size of the circle ).

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[@MathAndScience]

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[@qusaialhariri8117]

your explanation is really amazing and I actually have learned a lot from that lesson. Thanks a lot sir and keep it up

[@MathAndScience]

You are very welcome!

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[@English-ug2cs]

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[@MathAndScience]

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[@mathfullyexplained]

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[@haleymills239]

Just a question but I was told you minus exponents when dividing. So why instead of x/x^2 do you not get -x and have to move it up?

[@jonwill]

Jason, (or anyone) On the problem just before the last 31:08 with the result: x-2/x-4, why wouldn't you further factor x-4 as a difference of squares? Factoring as: (√x+2)(√x-2). Is there rationale for no longer factoring when factoring further is possible? In general does one stop factoring when no cancellations will result? Any insight is appreciated.

[@gtlasers1]

factoring further when no cancellations are possible isn't needed for simplifying rational expressions. Once you've factored to the point where no further simplifications or cancellations can occur, you've done enough and none else is needed. The expression x-2/x-4 does not have any common factors to cancel out, so it's considered fully simplified. Factoring x−4 into √ ( 𝑥 + 2 ) √ ( x ​ +2) wouldn't help because it isn't a good factorization for non-perfect square X. In summary, additional factoring is not required if it doesn't lead to cancellations or simplifications of the origin equation.

[@jonwill]

@@gtlasers1 Thank you for helping me understand why Jason stopped factoring, and the general rule for when to stop. I much appreciate your input.

[@jasminparveenparveen7061]

You explain very well sir 👌👌

[@MathAndScience]

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[@b.p.sadashivamurthy2575]

Hello sir, I have found the five rational numbers between 1 and 2 using equivalent fraction. The five rational numbers are - 7/6 , 8/6 , 9/6, 10/6 , 11/6. Sir, is it compulsory that we should simplify the above five rational numbers. Please help me with the above doubt. Thank you.

[@richardbrookens2896]

7/6, 4/3, 3/2, 5/3, 11/6

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[@shawnouellette1953]

Why can't you multiply and divide irrational expressions perfectly?

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Is this part of algebra 1 or 2 ?