5
votes
7answers
293 views

If $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$.

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
2
votes
2answers
44 views

Determine a quartic equation

I am working on a puzzle from Popular Science from the 1980's. This was a puzzle that existed before pocket calculators or programmable computers. It results in one needing to solve a seeming quartic ...
0
votes
3answers
49 views

How do I compute the individual terms of a polynomial to the power of -1?

If my polynomial $p$ is: $x+1$, obviously $p^{-1} = \frac{1}{x+1}$. Is it possible for me to split $\frac{1}{x+1}$ into a sum of two terms? In other words, is there an algorithm to write $p^{-1}$ as ...
0
votes
4answers
41 views

Recognizing the proper polynomial factorization to solve an indeterminate limit

I had to solve the $\lim_{x \to 3} \frac{x^3-3x^2-x+3}{x^2-x-6}$ that is indeed an indeterminate form ($\frac{0}{0}$). The approach I adopted was to factor the polinomials so that I can deviate from ...
6
votes
0answers
249 views

IMO 1979 problem

The question is $$If\, p,q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad Prove \,that\, 1979|p.$$ So my solution went like this: ...
3
votes
4answers
157 views

Can't simplify this fraction.

I've been having trouble simplifying this fraction : $$ \frac{1+x^6}{1+x^2} $$ Can anyone explain step by step on how to solve this? Thank you.
4
votes
2answers
79 views

Why does my intuition for “order of divergence” for algebraic fractions fail?

I come across this identity once in a while but I actually never grasped it: $$\frac{2}{1-x^2}=\frac{(1-x)+(1+x)}{(1+x)(1-x)}=\frac{1}{1+x}+\frac{1}{1-x}$$ I'm surprised by it because I would ...
1
vote
2answers
81 views

How can be done by the method of mathematical induction?

We are given that $P(x+1)-P(x)=2x+1$ We also know that $P(0)=1$ We want to prove that $P(2004)=(2004)^2 +1$ Can someone explain how can be solved with mathematical induction? Thank you in advance!
0
votes
2answers
67 views

system of equations with three equations.

We have to find all real solutions to this system of equations: $$x=\frac{4z^2}{1+4z^2},y=\frac{4x^2}{1+4x^2},z=\frac{4y^2}{1+4y^2}$$
3
votes
1answer
108 views

How to solve this System of Polynomial Equations?

I have to complete a summer packet of 90 Algebra 2 questions. I have completed 89 of them, the only one I could not get was this. I know the answer is $y = \frac {47}2$, $\frac 17$ according to ...
0
votes
1answer
118 views

Simplify this algebraic fraction

I have this algebraic fraction: $$\frac{t^4-1}{t^2-t^6}$$ And I'm told the answer is: $$\frac{-1}{t^2}$$ I can't for the life of me work out how to simplify it. (I'm sorry for the simple question) ...
0
votes
0answers
160 views

Partial fraction expansion when the degree of the numerator is unknown

Hope it's not too stupid: is there any general approach to partial fraction expansion when the degrees of polynomials in the numerator are unknown?