Simplifying Rational Expressions Part 1

This lesson shows how to simplify rational expressions. This is the first part of a two part lesson. This was a lesson created for the MCR3U Functions course in the province of Ontario, Canada.

[AlRichards]

The videos I upload are from the classes I teach. I use YouTube as an extra resource for my students. I don't have much spare time to make extra ones and they are time consuming. For that reason I don't take requests. Sorry.

[IkabobJr]

I CAN REWIND, PAUSE, AND AND TRY IT BEFORE HE DOES IT. I LOVE IT!! YOU TUBE TAKE OVER SCHOOL!!!

[sarahwhite]

You explain the laws/rules and functions more than Khan, thank you. Helping much more.

[skarties]

I love you, I have a exam today and it's all on this. I completely understand and I need a 100 A too keep me on a honor roll . I didn't wanna fail and you're helping me thank you!

[AlRichards]

If you divided 18x^3/12x by 6 (top & bottom) you would get 3x^3/2x, but there would still be a common factor of x that will divide out of the numerator & denominator. Hence the 3x^3/2x would reduce to 3x^2/2.
Instead of doing the above, it's quicker to divide the original 18x^3/12x by 6x to get 3x^2/2.

[AlRichards]

The x^2 isn't ignored in part D. x^2 + x - 12 factors into (x + 4)(x - 3). You can always check any factoring by expanding the factors together. Remember that x times x equals x^2. Hope this makes sense.

[AlRichards]

Examples a) & b) were both a monomial divided by a monomial. Starting in c) it's a binomial divided by a monomial (or larger later on). So in c) I common factored the x out of the numerator so it would divide out with the 4x in the denominator.

[ryashairuna]

Thank you for the tutorial, Sir. It helped me alot for my examination tomorrow. I've got limited time to study for a lot of subjects since I'm taking a special one. Thanks for all your efforts!

[AlRichards]

Well, this math teach doesn't hate algebra. Algebra is the language of mathematics. It is true that most people won't use algebra much in "real life", but much more than math teachers use it. Engineers, surveyors, even some construction peple use algebra. If you were carpeting a special room that was in the shape of a trapezoid and you wanted to find the area, what's easier to work with, the formula A = h(a + b)/2 or Area = height times (side 1 minus side2) divided by two-that's algebra too.

[AlRichards]

All the restrictions represent are any values that would make the expression undefined. For example, you cannot divide by 0 (since it's undefined) so any number that gives a zero value in the denominator is a restriction. For example in the d) question the (x + 4) factor divided out. Now if x = -4 then x + 4 has a value of 0 since -4 + 4 = 0. So if x = -4 then the expression has a value of 0 on the top and bottom, which make it undefined. So the restriction is x cannot = -4.

[jheka]

you're waaayyy better than my teacher .. thanks a lot !!

[AlRichards]

@BryanDavilaTv2 All the restrictions represent are any values that would make the expression undefined. For example, you cannot divide by 0 (since it's undefined) so any number that gives a zero value in the denominator is a restriction. For example in the d) question the (x + 4) factor divided out. Now if x = -4 then x + 4 has a value of 0 since -4 + 4 = 0. So if x = -4 then the expression has a value of 0 on the top and bottom, which make it undefined. So the restriction is x cannot = -4.

[AlRichards]

@c15j01 We study algebra because it is the language that we communicate when doing any math. Take a simple area formula from geometry. If we didn't have algebra then the area of a triangle, A = bh/2, would become "Area equals base times height divided by 2", or a formula for $100 compounded at 4% interest for 5 years (A = 100(1.04)^5) becomes "accumulated amount equals 100 times 1.04 to the power of 5". Algebra is like a game. Make sure you know the rules and it is a lot simpler.

[AlRichards]

@c15j01 Al-Khwarizmi wrote ... book that changed mathematics forever: "The Compendious Book on Calculation by Completion and Balancing." Two of the words in the Arabic title are al-jabr, from which we derive the English word algebra. It is the symbolic language on which much of higher mathematics is based.
[From Yahoo Answers]

[AlRichards]

You can only factor an x out of the 10x^2 - 13x because the 13x only has an x^1 to factor out. Also, even if you could have factored out a higher power of x in the numerator, you can only divide x^1 out of the denominator anyway, so factoring more out of the top wouldn't help simplify it anymore. So, if it was (10x^3 - 13x^2)/4x you could still only divide out an x and it would simplify to (10x^2 - 13x)/4

[joselesanroman]

OMG you explain so well!!! thank you for this tutorial, it was exactly what I needed!!

[AlRichards]

Factor the denominator to get 3ab/[6ab(a - b)] and then the 3ab divides into the 6ab factor in the denominator leaving a 2 there (since 6/3 = 2) so we are left with 1/[2(a - b)] or 1/ (2a - 2b). The top still has a 1 there since 3ab divided into 3ab = 1.

[AlRichards]

I'm assuming you are asking how you'd simplify (2x - 12)/(2x + 14). The only thing you can do here is common factor a 2 out of both the numerator & denominator to get 2(x - 6)/2(x + 7). After the 2s divide out it simplifies to (x - 6)/(x + 7).

[AlRichards]

Common factor each to get 2m^2n(n - 3)/[6mn^2(1 - 9n^2)], then you can divide out a 2mn out of the common factors to get m(n - 3)/[3n(1 - 9n^2)].

[AlRichards]

(42k^2 - 36k)/12k^2 = 6k(7k - 6)/12k^2 = (7k - 6)/2k after dividing a 6k out of the top and bottom.