Articles of algebra precalculus

Simplifying $\frac{\;\frac{x}{1-x}+\frac{1+x}{x}\;}{\frac{1-x}{x}+\frac{x}{1+x}}$

$$\frac{\;\;\dfrac{x}{1-x}+\dfrac{1+x}{x}\;\;}{\dfrac{1-x}{x}+\dfrac{x}{1+x}}$$ I can simplify this using the slow way —adding the numerator and denominator, and then dividing— but my teacher told me there is another way. Any help?

Prove that $2^n +1$ in never a perfect cube

Prove that $2^n +1$ in never a perfect cube I’ve been thinking about this problem, but I don’t know how to do it. I know that if $m^3=2^n+1$, then $m$ should be an odd number, but I ‘m not able to get to a contradiction.

Prove that $\sqrt5 – \sqrt3$ is Irrational

I’ve gone many directions and they all fail. The sum of two irrationals doesn’t need to be irrational. I found a proof saying: if irrational $x,y$ have a rational sum $x+y$, then $x-y$ is irrational, or vice versa. However, in this case $x+y$ and $x-y$ are irrational. I must have misinterpreted the proof maybe. Is […]

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is within $\varepsilon$ of an integer, is an $IP^*$ set. Being an $IP^*$ set is a notion of largeness/combinatorial richness. A set […]

Why is $i^i$ real?

Possible Duplicate: How to raise a complex number to the power of another complex number? My calculator (as well as WolframAlpha) gives me the approximation: $$0.2078795763507619085469…$$ But I don’t understand how exponentiating two purely imaginary constructs yields a real (albeit irrational) number. When I do $i^{i+1}$ it gives me an imaginary number as well as […]

Finding the zeroes of the function.

I would like to find the zeroes of the following function given that $3-i$ is a zero of $f$: $f(x) = 2x^4-7x^3-13x^2+68x-30$ Please explain to me how to do this problem. Thanks!

How to make four 7 s equal to 4 and to 10?

How to make four 7 s equal to 4 and to 10? f(7, 7, 7, 7) = 1: 7/7 * 7/7 = 1 f(7, 7, 7, 7) = 2: 7/7 + 7/7 = 2 f(7, 7, 7, 7) = 3: (7+7+7)/7 = 3 f(7, 7, 7, 7) = 4: ? . . f(7, 7, 7, […]

Advanced Algebraic Equation – Solve for P

Please solve the following equation and leave your answer in terms of P. This problem has been bugging me for months now, and I have not been able to reduce it past this form: $$ (P/(L-P))^K * ((P-K)/P)^L = Ae^({KT}(LK)(L-K)) $$ where $A = e^c$ and c is some constant. I have tried to solve […]

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can’t see where we are going to use the ‘strictly increasing’ fact.I don’t think there is a way to reverse-engineer such a function without heavy machinery.I have plugged in loads of […]

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}} \operatorname{d}t \tag8$ either by analytical method or from geometrical method with out using numerical methods? means looking for a closed form with out infinite series expansion in the result NB: Main issue is the lack of […]