how to setup partial fractions (all cases)

Calculus tutorial on how to set up partial fraction decompositions. We will cover all cases: distinct linear factors, quadratic factors, and repeated factors. Here's the video on how to solve for the constants: • how to solve partial f...
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[@bprpcalculusbasics]

How to solve the constants: kzbin.info/www/bejne/jaOqiXygjbeGr9U

[@waleedd857]

this, is HANDS DOWN, the BEST explanation on this entire website, maybe even in this entire universe

[@bprpcalculusbasics]

Thanks!!

[@lyndonmensah-cooley869]

the most captivating part is the way he switches between the 2 markers with one hand

[@willbishop1355]

This is great. I've done a lot of calculus and watched a lot of math videos, but don't know that I'd ever had it explained this way before.

[@bprpcalculusbasics]

Thank you.

[@leif1075]

@@bprpcalculusbasics Donyou think thisbis too unintuitive though and needlessly complicated? Hope you can respond when you can. Thanks for sharing.

[@martinezfalcon5217]

Finally this question is answered in an intuitive way:)) I’ve been wondering this for ages

[@luisaleman9512]

This is the best explanation I've seen for this. Great job.

[@farhansadik5423]

You actually don't know how much I appreciate this video, i've had this question looming over my head for like 3 years! This was actually astonishing to see! Thanks!

[@rodrigosilva893]

Same to me, I just love this guy 😭

[@farhansadik5423]

​@@rodrigosilva893you made me watch this again, thanks 🫂

[@bprpcalculusbasics]

Glad to help! Thank you!!!

[@stemwithme594]

This was so helpful, listing all the possibilities and literally answering all the questions just as I was asking myself!

[@vaishnavimajumdar561]

this was the best someone have ever explained me. It was to the point and not too overwhelming. By the way, that poke-ball mic is on point!!

[@wisdom6486]

One of Greatest Mathematics Teacher🤗😊

[@user-wu8yq1rb9t]

Thank you so much dear Teacher 💖

[@user-xx8zs3us7j]

I needed a quick reminder before exams thanks

[@ultimateboss5161]

Sup

[@JohnSmith-pv1jq]

I wish my university math teachers were like this guy!

[@BitsNBytes_]

This is incredibly helpful, thanks a lot man!

[@argghsgahghrw]

you explain math so comprehensively thank you

[@azmaeenadib3821]

thank you sir, it helped a lot.

[@caghey]

i finally understood thanks!!

[@user-rs1gs6vg3g]

Great video. I was most impressed by the slight of hand with the dry erase markers though

[@Bruh-bk6yo]

Good thing this answer got revealed during our algebra lecture when we were proving that every polynomial fraction can be expressed as the sum of the polynomial simple fractions.

[@fafa47t]

Wow so cool ! Tysm❤

[@cloud_222]

Thank you so much!!

[@ArtStarproductions]

That was awesome. Thank you! Helping my college teen!

[@user-wu8yq1rb9t]

The best way to do partial fraction, is not to do partial fraction! ... I wish, But it seems impossible! I'm always struggling with Partial fractions! Thank you Teacher

[@yasinchakmak1972]

same😁

[@AlfonsoNeilJimenezCasallas]

very useful to solve some kind of integrals and differential equations

[@aes_user]

Thank you so so much.This was very helpful

[@roman.r8587]

Thanks, super helpfull!!

[@marvinlwando9420]

U ARE A LIFE SAVER!!! THANKS

[@eftsa5323]

thank you very much my friend. Greeting from Greece!

[@user-kg8kq8sh3x]

Thanks I got it good.. best explanation

[@garv_g]

Another nice video...❤ I revise My concepts watching your videos...

[@bprpcalculusbasics]

Glad to hear that

[@user-yb1qx4jm8g]

Thank you so much that gave me a huge help.

[@kingbeauregard]

For me, the math is more intuitive this way, but it gets to the same place. Looking just at the (x + 2)^2 part, we could express its partial fraction as (Bx + C) / (x + 2)^2. Then we could algebrize that numerator into "(B(x + 2) + (C - 2B))", and since we're dealing with as-yet undetermined coefficients, we could replace "(C - 2B)" with "D". So that leaves us with a numerator of "B(x + 2) + D", and when we divide by "(x + 2)^2", we're left with "B/(x + 2) + D/(x + 2)^2".

[@dungdul4151]

That's so much simpler! The substitution thing he did in the video is too unintuitive and overcomplicates the problem in my opinion.

[@leif1075]

But WAIT can't you do the partial.fractuon decomposition WITHOUT the B term..I think you can so why doesn't he show this??

[@carultch]

​@@leif1075 You might have one that coincidentally turns out to be zero, but in general, if you are working with a quadratic, you have to put a linear term on top of it, because fractions using a quadratic denominator have two degrees of freedom. You can either set it up as (B*x + C)/(x + 2)^2, or B/(x + 2)^2 + C/(x + 2). Either way, you'll get a partial fraction decomposition that is valid. Usually, the form of B/(x + 2)^2 + C/(x + 2) is preferable, for an application where you'd use this concept. I recommend putting terms you can get by Heaviside coverup first. So if I were given (2*x + 1)/((x + 1)*(x + 2)^2), I'd assign A to be the coefficient over (x + 1), and B to be the coefficient over (x + 2)^2. Both of these, can be found with Heaviside coverup. You get A = -1, and B = 3. There is a shortcut that can work for getting C. (2*x + 1)/((x + 1)*(x + 2)^2) = -1/(x + 1) + 3/(x + 2)^2 + C/(x + 2) Multiply through by just one instance of (x + 2): (2*x + 1)/((x + 1)*(x + 2)) = -(x + 2)/(x + 1) - 3/(x + 2) + C Let x go to infinity by taking the limit. Terms with a higher x-degree in the denominator than the numerator become zero, terms with equal degrees of x in both become a finite number. 0 = -1 - 0 + C And we can directly see that C = +1.

[@leif1075]

@@carultch indont known what heaviside cover up is nut also what you say doesn't make sense..you already have one linear term.and one quadratic term so you still.dont need the B term..see what I mean??

[@carultch]

@@leif1075 Heaviside cover-up is a shortcut for partial fractions. The idea is that you can more directly find the coefficient over the linear terms, by plugging in the input that makes that particular factor equal zero. Then, cover up that term in the original fraction, and evaluate the rest at the same input value. If you try only having one constant coefficient over either a quadratic denominator, or a repeated linear denominator without the second term, you'll end up with an over-constrained system of equations, that is either redundant or contradictory. Try it for: (2*x + 1)/((x + 1)*(x + 2)^2) Assume it is equal to: A/(x + 1) + B/(x + 2)^2 Cross-multiply: (2*x + 1) = A*(x + 2)^2 + B*(x + 1) Expand: 2*x + 1 = A*x^2 + 4*A*x + 4*A + B*x + B This constructs the equations: A = 0 4*A + B = 2 B = 1 And you'll see a contradiction, that B needs to both equal 1 and 2, which it can't do. This is why we need a third term, of C/(x + 2), so we have a third unknown for a 3-equation system.

[@ammarns8858]

For more tutorials 😂😂 you know la kho haha. Thanks

[@evanbarkman5786]

I disagree that that's the reason, it's definitely a good way to show it, but the reason is because of what partial fractions are, essentially they are undoing finding the common denominator, and when you have a repeated term, you could have had every power of the repeated term up to the power of the term and the denominator would have looked the same, so you assume they could all be there, unless you show a numerator is zero. My biggest beef with partial fractions is when I first learned how to do them I didn't learn why I was doing them, just that it was how you solved a certain type of integral, and therefore I didn't really learn how to do it properly, now that I understand that you're just reversing getting a common denominator it all makes sense, and I could probably figure it out from just the idea if I really wanted to, rather than having to remember the process (although it would be a little faster if I could remember it).

[@bprpcalculusbasics]

"essentially they are undoing finding the common denominator, and when you have a repeated term, you could have had every power of the repeated term up to the power of the term and the denominator would have looked the same, so you assume they could all be there, unless you show a numerator is zero." okay but why is this the case?

[@dogfrog3893]

@@bprpcalculusbasics they’ve been real quiet after this.😂

[@dVPulse]

@@bprpcalculusbasics I understood all of the video except the rule you used; the degree on the top must be 1 less than the degree on the bottom. Why is this the case?

[@chessematics]

@@dVPulse that requires some proof which can't be written down in the comments without making its too tiresome to read. But yeah, there IS a proof and I'll post a link whenever i find it.

[@nvx8408]

@@chessematics where?pls post it

[@KrishVashist-gn7il]

This guy is just great!!

[@davidolosunde5046]

Thanks sir, I love your teaching

[@theofelder6568]

thank you, very helpful!

[@d9sign]

Boom, I was searching for long time

[@OdetaZenelaj]

u helped me soo much❤

[@DeJay7]

Just today I watched a video where partial fractions were used, insane timing.

[@satyendranooka3437]

exactly the video I wanted ! 😂 #NCERT,Class 12

[@user-cc4lj3ge5u]

The easiest maths book you could ever do 😂

[@Mariosergio61]

A big class. thanks.

[@bprpcalculusbasics]

Glad you liked it!

[@mateomentor]

Thank you so much!

[@blug2949]

This helped me so much becuse of this i will be able to pass my precalc class helping me raise it by over TWO percent!!! My parents are finally satisfied.

[@codexe.]

what an absolute legend

[@evin4899]

Very Great Video

[@JinkunYan]

it is a really good way to understand it. You forget what we need to prove at 8:40, HHHHA~! But I got it, thank you so much.

[@user-su9wz4jv3f]

Thank you🎉

[@riverevr6891]

Good stuff

[@starzfn9109]

I have a problem for you.What are the angles at the intersection point of the functions (1/3)^x and 2^x.I found this problem really fun to do 😄

[@NoNameAtAll2]

arctg(ln(3))+arctg(ln(2))? does it simplify somehow or smth?

[@user-dl8nk5bf8v]

@@NoNameAtAll2 Setting a := arctan( ln 2 ), b := arctan( ln 3 ) we have tan a = ln 2, tan b = ln 3, tan( a + b ) = ( tan a + tan b )/( 1 - ( tan a )( tan b ) ) = ( ln 2 + ln 3 )/( 1 - ( ln 2 )( ln 3 ) ) = ( ln 6 )/( 1 - ( ln 2 )( ln 3 ) ). The calculator shows that ( ln 2 )( ln 3 ) = 0.7615000… So tan( a + b ) > 0. Noting that 0 < a, b < π/2, 0 < a + b < π, we have a + b = arctan( ( ln 6 )/( 1 - ( ln 2 )( ln 3 ) ) ). Consequently, we obtain arctan( ln 2 ) + arctan( ln 3 ) = arctan( ( ln 6 )/( 1 - ( ln 2 )( ln 3 ) ) ).

[@user-dl8nk5bf8v]

In general, for every x, y > 0 we have ( i ) 0 < xy < 1　⇒　arctan x + arctan y = arctan( ( x + y )/( 1 - xy ) ) ( ii ) xy = 1　⇒　arctan x + arctan y = π/2 ( iii ) 1 < xy　⇒　arctan x + arctan y = π - arctan( ( x + y )/( xy - 1 ) )

[@NoNameAtAll2]

@@user-dl8nk5bf8v idk man having _fraction_ with logarithms doesn't seem simpler than 2 separate ones and arctg layer is still there, so it's cubersome either way

[@user-dl8nk5bf8v]

@@NoNameAtAll2 Oh, sorry. I don't know what expression is better for some calculation.

[@logenlogen9267]

Thank you 🎉🎉🎉🎉

[@lame77151]

i love you man

[@4Shams]

I need a specified video for how to solve partial fractions, pls🙏

[@shahzaibkhan9992]

Brilliant...

[@adityanagesh9554]

Useful for maths 2B integration.

[@TheLabbening]

Can we just admire how well he switches the markers xD

[@kepler4192]

blackpenredpen fans for the win! also a question, if you include complex numbers, would partial fractions be any different other than the irreducible being reducible?

[@carultch]

Yes indeed. If you include complex numbers, then all polynomial denominators up to the quartic, would technically be reducible. Quintics and beyond, have no closed-form solution for the roots, in elementary functions. It usually won't help you very much to do this. It's a lot easier to set up the algebraic system to solve for unknown coefficients, than to detour to the complex numbers. Even though Heaviside coverup works for linear factors of complex roots as well.

[@ajayiolajide6390]

Thanks a lot for this 'tutorial'😅. You could have chose your L1 but thanks for using the English language. I really appreciate it.

[@jeffreylin235]

That's the best explanation of this question.

[@jccab]

KING

[@user-dl8nk5bf8v]

Hi, I comment here for the first time. By Euclidean algorithm we have ( x + 2 )^2 = ( x + 3 )( x + 1 ) + 1, 1 = ( x + 2 )^2 - ( x + 3 )( x + 1 ), 1/( ( x + 1 )( x + 2 )^2 ) = 1/( x + 1 ) - ( x + 3 )/( x + 2 )^2. We also have ( x + 3 )/( x + 2 )^2 = ( x + 2 + 1 )/( x + 2 )^2 = 1/( x + 2 ) + 1/( x + 2 )^2. Hence we obtain 1/( ( x + 1 )( x + 2 )^2 ) = 1/( x + 1 ) - 1/( x + 2 ) - 1/( x + 2 )^2.

[@user-dl8nk5bf8v]

By the similar way we have 1 = ( x + 2 )^2 - ( x + 3 )( x + 1 ), 2x + 1 = ( 2x + 1 )( x + 2 )^2 - ( 2x + 1 )( x + 3 )( x + 1 ) = ( 2( x + 1 ) - 1 )( x + 2 )^2 - ( 2( x + 2 )^2 - ( x + 5 ) )( x + 1 ) = ( x + 5 )( x + 1 ) - ( x + 2 )^2, ( 2x + 1 )/( ( x + 1 )( x + 2 )^2 ) = ( x + 5 )/( x + 2 )^2 - 1/( x + 1 ). And we also have ( x + 5 )/( x + 2 )^2 = 1/( x + 2 ) + 3/( x + 2 )^2. Hence we obatin ( 2x + 1 )/( ( x + 1 )( x + 2 )^2 ) = 1/( x + 2 ) + 3/( x + 2 )^2 - 1/( x + 1 ).

[@Yazeed696]

before watching the video, ik i am going to understand it babe, lets gett itttt. midterm 3 tomorrow bout to ace it with only 6 hours of studying

[@bprpcalculusbasics]

Best of luck!!!

[@user-ue7tn5yz3z]

thank you bzzzzaf

[@ayushman3339]

W explanation.

[@vivada2667]

NOO WHERE ARE THE MORE TUTORILS 😭😭

[@YunusTalhaOzsahin]

Legend

[@hinochii4284]

Thank you for answering my biggest WHY .. I asked my professor about it and he didn't give me a clear answer.

[@patricknazar]

Tried to get the answer to this question, it was nowhere to be found. Then all of a sudden this appears on my home screen. Wow. Thanks

[@bprpcalculusbasics]

YT read your mind! Cheers : )

[@raiyanreza9764]

sorry but this was so funny 🤣 @06:34 but kudos for the amazing explanation

[@jonold975]

Understanding the raising power rule is better than memorizing most of equation😂😂

[@ZinhlePercy]

the end caught me off guard🤣🤣🤣🤣

[@ZaynShaheen]

Thank you man , you're great 👏 And the and was like u can't forget ur Asian language 🤣💚

[@DragonflyLover618]

I love you.

[@dhukurpanivlogs3405]

Same type of question asked in my math exam

[@mohamedibrahim1023]

Very nice video tho but why we need the numerator has to be only one degree less than the denominator?

[@bprpcalculusbasics]

Bc if not, then we can do polynomial long division.

[@pto314]

Try to split as A(x+1) + B(x+1), and you’ll see why we need the 2nd denominator as (x+1)^2.

[@alexdemoura9972]

I like it: gold on black golden ratio. Maybe a coincidence, these last weeks I am working on statistical techniques to be applied on Logistics, Demography, Approximation, Politics (solutions for Gerrymandering) and so on... golden ratio has a fundamental role in these techniques, as an alternative number for Nature and Human objects which dimensions have unknown statistical distribution.

[@deenreviews]

Dude I love you, you are gonna save me from getting absolute smoked, obliterated, cooked, and bombarded by this math exam

[@gregoryoruko]

Notice the t-shirt is printed golden ratio number

[@GF86123]

This is now more confusing than before I watched this video!!

[@vincenzopeiranosanchez3938]

I love you

[@birukdamtew]

IV. If (a * x ^ 2 + bx + c) ^ k , for a ne 0and * b ^ 2 - 4ac < 0 , is a factor in the denominator for k > 1 and (a * x ^ 2 + bx + c) ^ (k + 1) not a factor, then the corresponding sum of partial fractions to this factor is: (A_{1}*x + B_{1})/(a * x ^ 2 + bx + c) + (A_{2}*x + B_{2})/((a * x ^ 2 + bx + c) ^ 2) +***+ A k x+B k (ax^ 2 +bx+c)^ k , where A_{1} ,...,A k and B_{1} ,...,B k are constants that we have to determine. I've seen a lot of videos but can't find this thing except my text book.

[@munshatmuhtadee2013]

What if t is a second degree expression? If we follow the same process,we'll encounter sqrt in the denominator which will not be good to work with.

[@mathisnotforthefaintofheart]

It can be done without the repeated factor of (x+2). Just write A/(x+1) with (Bx+c)/(x+2)^2. You get a system of three equations in 3 unknowns which is easily solvable.

[@TheHansaj]

Very helpful concept. i'd do you a favor- Say it like, "two toe real".

[@MonkOrMan]

But why does degree of the numerator have to be 1 less than that of the denominator? Your explanation relies on this but you don't explain why. Other than that it's a great video! I'm glad that people like you are here to explain WHY we do things rather than just how to do them like most teachers and youtubers

[@aewcontrol2984]

Perhaps because you could use long division instead.

[@carultch]

If the denominator is greater than or equal to the numerator, then there are other ways of simplifying the fraction, to prepare them for integration or inverse Laplace transforms. The numerator doesn't have to be just 1 less than the denominator, just that it is less than the denominator in general. As a couple examples, consider: (x^2 + 2*x + 4)/(x + 1) With synthetic division, it will reduce to: x + 1 + 3/(x + 1) Only the 3/(x + 1) needs to be integrated with methods of integrating algebraic fractions. The x + 1 part can simply be integrated with the power rule. As another example, that has an equal power to its denominator, consider: (x^2 + 2*x + 4)/(x^2 + 5*x + 6) For this one, we can add zero in a fancy way, to separate it down to a polynomial expression, and a rational expression. x^2 + 2*x + 4 + (3*x - 3*x) + (2 - 2) = x^2 + 5*x + 6 - 3*x - 2 This allows us to rewrite it as: (x^2 + 5*x + 6)/(x^2 + 5*x + 6) - (3*x + 2)/(x^2 + 5*x + 6) And ultimately reduce it to: 1 - 7/(x + 3) + 4/(x + 2) So the fact that there were equal numerator and denominator orders, just means that there is a polynomial term out in front, followed by a fraction where the numerator order is at least one less than the denominator.

[@athinrahman1902]

Guy got great skills with his marker lol

[@TRak598]

The way I rationalize repeating factors is that you need the possibility of getting different values for A, B, C, etc, as a repeating factor is but a representation of a second degree equation. However, we only have one value for all the roots of such equations (delta is 0), hence you need to have a way to generate the other factors. Since the degree of repetition determinates the number of equal roots to the equation, you can use every exponential combination, and they will alow you to make MMC since the equation is divisible by all the degrees from 1 to its actual degree. TL; DR: It's not like I can't write B / (x-a)^n and call it a day. It's just that B won't be a single number in this case, so instead of writing it that way and then doing partial fraction stuff to it again, we skip directly to writing all the different factors with different powers of the same (x-a) bit as the divisors.

[@Its_Juss_Lithas_channel]

Its the end for me 😂... for more tutorias

[@kotjiiherunga4764]

😂😂😂

[@dan_mirnejhad]

why does the degree on the top have to be one less than the degree on the bottom when setting up the partial fractions?

[@bprpcalculusbasics]

Bc if not, then we can do long division to break it down

[@carultch]

If the degree on top, is equal to, or greater than, the degree on bottom, then you can simplify the rational expression, into a separate polynomial added to a fraction. Either by adding zero in a fancy way, to form a term you can cancel, or by using polynomial division. The polynomial terms on their own, once extracted from the fraction, can be handled with the power rule. As for why it is one less, rather than two less, the reason is that you'll get an indeterminate system of equations when attempting to solve for coefficients, if you don't have a big enough polynomial on top of any given term. You could "get lucky" and end up with a highest degree term of the numerator with a coefficient of zero, but in general, you need at least a polynomial on top of a degree one less, than the degree of the polynomial on bottom.

[@carultch]

As an example, given: x^3/[(x + 1)*(x + 2)] I can simplify this to: x - 3 + (7*x + 6)/[(x + 1)*(x + 2)] Once in this form, the x - 3 part, is all set for calculus applications. It's just the power rule to either differentiate or integrate it. All that remains for partial fractions is: (7*x + 6)/[(x + 1)*(x + 2)] This one is a proper fraction, with a linear term on top, and a factored quadratic on bottom. The simplified expression after partial fractions is complete, is: x - 3 - 1/(x + 1) + 8/(x + 2)

[@claire.c]

4:09 where the biggest why is answered. Also, why does the numerator have to be at least 1 power less than the denominator?

[@kobethebeefinmathworld953]

For more tutorials, (video ends) 😅

[@tedle2948]

Thank you sir, but how to define what is a linear ?

[@user-cc4lj3ge5u]

What happens when two irreducible quadratic are in the denominator ? For example x/[(x^2+x+1)(x^2+1)]

[@carultch]

If you have two irreducible quadratic denominators, then you have two separate irreducible quadratic terms, each with two unknown coefficients forming linear numerators. For your example: x/[(x^2+x+1)(x^2+1)] I recommend completing the square first, when you have an x-term in the middle. Most applications of partial fractions, will eventually require completing the square. x^2 + x + 1 = (x + 1/2)^2 + 3/4 Set up the partial fractions, with each quadratic denominator, and two arbitrary linear numerators: (A*x + B)/(x^2 + 1) + (C*(x + 1/2) + D)/((x + 1/2)^2 + 3/4) Note that since we have an offset quadratic for the second term, instead of using C*x + D, I've opted for C*(x + 1/2) + D instead. You'll see why. Equate to original expression: (A*x + B)/(x^2 + 1) + (C*(x + 1/2) + D)/((x + 1/2)^2 + 3/4) = x/[(x^2+x+1)(x^2+1)] Plug in strategic values for x, to solve for the coefficients. Usually 1 and 0 are strategic values to use, since they keep your math simple. In this case, -1/2 is also a strategic value. I'll also use x=-1. Be careful with x=-1 and x=+1, since they could give redundant information. You can also use x=infinity in some cases, though it won't help us here. For x = 0, this makes the A disappear: (A*0 + B)/(0^2 + 1) + (C*(0 + 1/2) + D)/((0 + 1/2)^2 + 3/4) = 0/[(0^2+0+1)(0^2+1)] B + C/2 + D = 0 For x = -1/2, this makes C disappear: (-1/2*A + B)/((-1/2)^2 + 1) + (C*0 + D)/((-1/2 + 1/2)^2 + 3/4) = -1/2/[((-1/2)^2 - 1/2 + 1)((-1/2)^2+1)] -4/10*A + B*4/5 + 4/3*D = -8/15 -6*A + 12*B + 20*D = -8/3 For x = 1: (A*1 + B)/(1^2 + 1) + (C*(1 + 1/2) + D)/((1 + 1/2)^2 + 3/4) = 1/[(1^2+1+1)(1^2+1)] A/2 + B/2 + C/2 + D/3 = 1/6 3*A + 3*B + 3*C + 2*D = 1 For x = -1: (-A + B)/(1 + 1) + (-C/2 + D)/((-1 + 1/2)^2 + 3/4) = -1/[((-1)^2+-1+1)((-1)^2+1)] -A/2 + B/2 -C/2 + D = -1/2 -A + B - C + 2*D = -1 Solve system: A = 0; B = 1; C=0; D=-1 Solution: 1/(x^2 + 1) - 1/((x + 1/2)^2 + 3/4)

[@logenlogen9267]

❤❤❤❤❤❤❤

[@normalhuman3569]

(a,b,c) = (-1,1,3)

[@alberteinstein3612]

The BIGGEST why: why go through the suffering? 😂 Especially when you have quadratic factors in the denominators so you need to convert them into arctan fractions

[@bprpcalculusbasics]

😂

[@politicalwrong3289]

3:36 But why degree in the numerator must be one less than degrees in the denominator?

[@mathisnotforthefaintofheart]

Because x^2+2 vis not factorable over the reals and so a linear factor is allowed to be a remainder

[@politicalwrong3289]

Thank you, this helped me a lot! ^_^@@mathisnotforthefaintofheart

[@gabrielcoimbra5850]

why the numerator has to be one degree lower than the denominator?

[@carultch]

It just needs to be less than in general, not necessarily only 1 less than. It could be 3 degrees lower than the denominator. If it is greater than or equal to the denominator's degree, there are two options we could take to simplify it: 1. Polynomial division, whether long division or synthetic division 2. Adding zero in a fancy way, so we can make our numerator's largest term, into a form we can reduce to a constant with the denominator. See this example: Given: (x^2 + 2*x + 4)/(x^2 + 5*x + 6) For this one, we can add zero in a fancy way, to separate it down to a constant, and a rational expression. x^2 + 2*x + 4 + (3*x - 3*x) + (2 - 2) = x^2 + 5*x + 6 - 3*x - 2 This allows us to rewrite it as: (x^2 + 5*x + 6)/(x^2 + 5*x + 6) - (3*x + 2)/(x^2 + 5*x + 6) And ultimately reduce it to: 1 - 7/(x + 3) + 4/(x + 2)

[@willie333b]

Ramanujan

[@bprpcalculusbasics]

?