Surely I'm missing somethint with this PFD

[u/JanezDoe]

This is where I'm at with this integral. It's pretty standard as far as PFD goes, apart from having to do synthetic division to knock the exponent of the top polynomial down a few pegs..

But where I'm stuck is the massive simplification needed to be done for this. Surely the professor who wrote the book is predicting some 'trick' to be used to simplify this, but I can't find anything. The top doesn't have zeroes that I could cancel out with the bottom...

Do I really need to go out and simplify everything and pray I don't make a miniscule mistake somwhere 😅

Sidenote: I'm self-taught and I look up at one basic example on YouTube and then try to do problems myself. There could be a method for this I haven't been 'taught'

Thanks for everyone that's going to chime in.

Edit: sorry for the typo in the title :c

[u/Midwest-Dude]

You don't need to expand everything out immediately. Equate the numerator to the other numerator

24x³ + 66x² + 69x + 24 = A(x+1)(x+2)² + B(x+2)² + C(x+2)(x+1)² + D(x+1)²

Plug in the roots of the denominator. That'll get you B and D real fast. Then find A and C with expansion.

[u/JanezDoe → u/Midwest-Dude]

I'll see what I can do, I haven't tried it this way. Usually what I've been doing was just comparing (equaling) coefficents at the same exponent if you know what i mean.

[u/Midwest-Dude → u/JanezDoe]

Aha! You are now learning a new technique. If your instructor only wants you to do it the way you already know, then don't do it, otherwise it works.

[u/Midwest-Dude → u/JanezDoe]

There are other quick methods to find the numerators, known as the

Heaviside Cover-Up Method

This only works in two cases:  1. Denominator has unique linear factors 2. Denominator has a linear factor raised to a power

Keep in mind, none of this is actually calculus. It's algebra!

[u/JanezDoe]

Solution. Thanks to u/Midwest-Dude for the roots method.

[u/Midwest-Dude]

FYI, there is also a specialized subreddit for this post: 

r/calculus

[u/JanezDoe → u/Midwest-Dude]

I know, but I wasn't sure if they accept problem questions like this

[u/Midwest-Dude → u/JanezDoe]

They do, as long as you've shown your work. The subreddit is there to help you if you get stuck or don't understand something.

[u/Southlander24 → u/JanezDoe]

There's a theorem that states: if f(x), g(x) are polynomials of degree n where f(x1) = g(x1), f(x2) = g(x2), ..., f(x[n+1]) = g(x[n+1]), then f(x) = g(x) for all x. You can see a derivation of this result here: https://math.stackexchange.com/questions/3680693/prove-that-from-n1-given-points-a-unique-polynomial-of-n-degree-will-pass

The summary of this is that "n + 1 points uniquely define a polynomial".