2
votes
1answer
80 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
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 ...
2
votes
1answer
71 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$
2
votes
4answers
209 views

How to find $\sqrt{1+{4\over x}+{4\over x^2} }$?

If $$abx^2 = (a-b)^2(x+1)$$ then what is $$\sqrt{1+{4\over x}+{4\over x^2} }$$ (A) $a+b \over a-b$ (B)$a-b\over a+b$ (C) $a^2+ab$ (d) None EDIT: What I've done is this: ...
2
votes
1answer
58 views

How to find $x + y + z$?

Q. If $x^{1/3} + y^{1/3} + z^{1/3} = 0$, then (A) $x + y + z = 3 xyz$ (B) $x + y + z = 0 $ (C) $( x + y + z)^3= 27 xyz$ (D)$ x^3 + y^3 + z^3 = 0$ What I've done: ...
1
vote
0answers
522 views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
9
votes
2answers
288 views

Algebraic equation problem - finding $x$

$$(x^2 +100)^2 =(x^3 -100)^3$$ How to solve it?
3
votes
0answers
73 views

Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
0
votes
2answers
96 views

How to efficiently compute the coefficients in a bi-binomial expansion?

Is there a computationally efficient way of calculating the coefficients of the polynomial expansion of expressions like $(1+x^a)^m(1+x^b)^n$ for arbitrary positive integers $m,n,a,b$ (and especially ...
1
vote
2answers
96 views

Understanding $\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$

I'm going through a book about algorithms and I encounter this. $$\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$$ How is this equation formed? If a theorem has been applied, what theorem ...
1
vote
1answer
145 views

Computing the power of real algebraic numbers

I'm looking for an efficient algorithm to compute the $n$-th power $\alpha^n$ of a real algebraic number $\alpha$ given by an interval representation for $n \in \mathbb{N}$. An interval representation ...
3
votes
2answers
395 views

How to compute coefficients in Trinomial triangle at specific position?

I need to compute coefficients of $i$-th power of $x$ from simplifying of $$(x^2 + x + 1)^n$$ From that site i know about trinomial triangle. But how can compute coeficients of $i$-th element at ...