Can anyone explain this problem on Partial fraction

Question

(CAUTION- PLEASE DO NOT TAKE MY QUESTIONS VERY SERIOUSLY.

I received a ban from asking questions,I don't know what to say really,I am just a student trying to learn,not a professional mathematician,so of course the questions could have been,well weird or useless to the mathematical community...I really did not mean any harm or anything except just trying to understand something.I really wish a feature like,telling you exactly what to do with your questions so the ban would get lifted was there.But I guess this is just hard luck and me not taking this community very seriously.I do apologize to the community,and I do understand that asking a question is a privilege not a right here.because,like to stop wasting people's time and not spread wrong ideas?right? I REPEAT,I DID NOT MEAN ANY CONFUSION OR ANYTHING,I AM NO PROFESSIONAL,JUST A LEARNER.)

Can anyone explain to me how the first part of the solution of the problem in the screenshot was arrived at?how did they express F(x) as a product of those two complicated polynomials?

Answer 1

After polynomial long division you must make the ansatz $$\frac{A}{x-2}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{Dx+E}{x^2+6}$$ For you computuation,the result is given by: $$1+{\frac {41}{10\,x-20}}+1/10\,{\frac {-x-2}{{x}^{2}+6}}-3\, \left( x- 2 \right) ^{-2}+2\, \left( x-2 \right) ^{-3} $$

Answer 2

This is how integer division (with remainder) works: dividing $n$ by $d$ we get

$$ n = q d + r $$

where $q$ is the quotient and $r$ is the remainder, $0 \leq r < d.$

Division of polynomials is analogous, but it is the degree of the remainder that must be less than that of the divisor. Dividing $N(x)$ by $D(x)$ we get

$$ N(x) = Q(x) D(x) + R(x) $$

where $0 \leq \deg R < \deg D$. It is also true that $\deg Q + \deg D = \deg N$ because the degree of the product of polynomials is the sum of their degrees, and since $\deg Q + \deg D > \deg D$ it is $Q(x) D(x)$ and not $R(x)$ that determines the degree of $N(x).$

So just from basic facts about polynomial division, we knew the first factor (which turns out to be $x + 2$) had to be a polynomial of degree $1,$ and we knew the remainder had to be of degree $3$ or less, therefore it can be written $r_3 x^3 + r_2 x^2 + r_1 x + r_0.$ The only fact that doesn't come directly from these facts about about $ N(x) = Q(x) D(x) + R(x) $ is that the author figured out that the leading coefficient of $Q(x)$ was $1$ before writing the form of $Q(x).$