Expert Answers to Algebra 2 Problems

Find the monthly house payments necessary to amortize a 6.0% loan of $234,700 over 25 years.

What is a formal model for equations in non-commutative ring?
The standard formal modeling for polynomials is the polynomial ring $R[X_1,...,X_n]$ which is a monoid ring R[Nn] over an rng R.
Under this construction, it is possible to commute $X_i$'s so that $X_i{X}_j=X_j{X}_i$.
However, over a non-commutative ring, we need more than this.
For example, say we want to find a solution $(x_1,x_2,x_3)$ of a equation $a{X}_1{X}_2^2{X}_3+b{X}_3{X}_1{X}_2=0$ over a non-commutative ring.
If we want to find 3-tuples in the center of the ring, then the standard polynomial ring can be used to find the solutions. However, in general, we don't want $X_i$'s to commute. For example, we want $(X_1{X}_2^2{X}_3)(X_3{X}_1)=X_1{X}_2^2{X}_3^2{X}_1$. Moreover, if there is a formal model for polynomials, it always form a ring.
Consquently, rather than ${\mathbb{N}}^n$, we need somewhat more complex structure.
Say, R[M] is a monoid ring over R and it is a correct model of polynomial rings.
Then, M should have information about how variables are ordered and what the powers of each variable in a term are. What would M be?
Is there a way to construct polynomials in non-commutative rings?

Are there more convenient ways of getting the number of digits of a positive integer?
I want to define nth power of 10 for a positive integer. Say for 43 it would be 2, for 5 it would be 1, for 9999 it would be 4. As for 1, 10, 100, ... I am still shifting between two options. 10 would result either 1 or 2. But for simplicity I select it to be 2.
In other words I want to get the number of digits of a positive integer with proper mathematical notation.
So far I have managed to do it with log and round:
$$\mathrm{floor}<!--\; \; -->({\log }_{10}<!--\; \; -->(43))+1=2$$
I'm just not sure if this is good approach. As a bonus it would be nice to have a solution working with any base, not just 10.

Is it possible for an irreducible polynomial with rational coefficients to have three zeros in an arithmetic progression?
Assume that $p(x)\in <!--\; \in \; -->\mathbb{Q}[x]$ is irreducible of degree $n\ge <!--\; \ge \; -->3$.
Is it possible that p(x) has three distinct zeros ${\mathrm{\alpha <!--\; \alpha \; -->}}_1,{\mathrm{\alpha <!--\; \alpha \; -->}}_2,{\mathrm{\alpha <!--\; \alpha \; -->}}_3$ such that ${\mathrm{\alpha <!--\; \alpha \; -->}}_1-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_2={\mathrm{\alpha <!--\; \alpha \; -->}}_2-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_3$?
As also observed by Dietrich Burde a cubic won't work here, so we need $\mathrm{deg}<!--\; \; -->p(x)\ge <!--\; \ge \; -->4$. The argument goes as follows. If $p(x)=x^3+c_2{x}^2+c_{1}x+c_0$, then $-<!--\; -\; -->c_2={\mathrm{\alpha <!--\; \alpha \; -->}}_1+{\mathrm{\alpha <!--\; \alpha \; -->}}_2+{\mathrm{\alpha <!--\; \alpha \; -->}}_3=3{\mathrm{\alpha <!--\; \alpha \; -->}}_2$ implying that ${\mathrm{\alpha <!--\; \alpha \; -->}}_2$ would be rational and contradicting the irreducibility of p(x).
This came up when I was pondering this question. There the focus was in minimizing the extension degree $[\mathbb{Q}({\mathrm{\alpha <!--\; \alpha \; -->}}_1-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_2):\mathbb{Q}]$. I had the idea that I want to find a case, where ${\mathrm{\alpha <!--\; \alpha \; -->}}_1-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_2$ is fixed by a large number of elements of the Galois group $G=\mathrm{Gal}<!--\; \; -->(L/\mathbb{Q})$, $L\subseteq <!--\; \subseteq \; -->\mathbb{C}$ the splitting field of p(x). One way of enabling that would be to have a lot of repetitions among the differences ${\mathrm{\alpha <!--\; \alpha \; -->}}_i-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_j$ of the roots ${\mathrm{\alpha <!--\; \alpha \; -->}}_1,\dots <!--\; \dots \; -->,{\mathrm{\alpha <!--\; \alpha \; -->}}_n\in <!--\; \in \; -->\mathbb{C}$ of p(x). For the purposes of that question it turned out to be sufficient to be able to pair up the zeros of p(x) in such a way that the same difference is repeated for each pair (see my answer).
But can we build "chains of zeros" with constant interval, i.e. arithmetic progressions of zeros.
Variants:
- If it is possible for three zeros, what about longer arithmetic progressions?
- Does the scene change, if we replace Q with another field K of characteristic zero? (Artin-Schreier polynomials show that the assumption about the characteristic is relevant.)

Why can't you add a coefficient before the logarithm base change rule?
Why is it that the rule
$${\log }_b<!--\; \; -->(x)=\frac{{\log }_c<!--\; \; -->(x)}{{\log }_c<!--\; \; -->(b)}$$
(the logarithm base change rule) is true but
$$a{\log }_b<!--\; \; -->(x)=\frac{a{\log }_c<!--\; \; -->(x)}{a{\log }_c<!--\; \; -->(b)}$$
isn't?
For example why does the equation,
$${\log }_{49}<!--\; \; -->3=\frac{\log <!--\; \; -->49}{\log <!--\; \; -->3}$$
work but
$$4{\log }_{49}<!--\; \; -->3=\frac{4\log <!--\; \; -->49}{4\log <!--\; \; -->3}$$
does not?
All you are doing is multiplying the logarithm by a term, and still multiplying each part by it when "splitting the logarithm up."
P.S. If the answer could be explained in somewhat simple terms it would be appreciated. I'm not like a math major or anything. :)

Question regarding logarithms 2
What is $\ln <!--\; \; -->(-<!--\; -\; -->1)$? And would there a taylor series for
$$\ln <!--\; \; -->\frac{1+x^m}{1-<!--\; -\; -->x^m}$$?

The point P is on the unit circle. Find P(x, y) from the given informatie The x-coordinate of P is $$\frac{12}{13}$$, and the y-coordinate is negative.

John wants to buy a new sports car, and he estimates that he'll need to make a $3,025.00 down payment towards his purchase. It he has 34 months to save up for the new car how much should he deposit into his account if the account earns 3.88% compounded continuously so that he may reach his goal?

Show that any invertible matrix has a logarithm.
I was trying to remember how to show that any invertible matrix has a (possibly complex) logarithm. I thought what I came up with was kind of cool, so I thought I'd post my answer here.

Why is modeling the joint distribution between many continuous random variables, obtains generalization more easily?
A fundamental problem that makes language modeling and other learning problems difficult is the curse of dimensionality. It is particularly obvious in the case when one wants to model the joint distribution between many discrete random variables (such as words in a sentence, or discrete attributes in a data-mining task). For example, if one wants to model the joint distribution of 10 consecutive words in a natural language with a vocabulary V of size 100,000, there are potentially $10000010-<!--\; -\; -->1=1050-<!--\; -\; -->1$ free parameters. When modeling continuous variables, we obtain generalization more easily (e.g. with smooth classes of functions like multi-layer neural networks or Gaussian mixture models) because the function to be learned can be expected to have some local smoothness properties. For discrete spaces, the generalization structure is not as obvious: any change of these discrete variables may have a drastic impact on the value of the function to be estimated, and when the number of values that each discrete variable can take is large, most observed objects are almost maximally far from each other in hamming distance.
Can someone explain, in simple words, why is that "When modeling continuous variables, we obtain generalization more easily"?

5 points A ball is dropped from a height of 12 feet. Each time it hits the ground, it bounces half of its previous height. What is the total distance travelled by the ball?

When does a polynomial f(x) generate an arithmetic sequence for consecutive values of integer x?
Given polynomial $f(x)=a_0+a_{1}x+\cdots <!--\; \cdots \; -->+a_n{x}^n,a_n\ne <!--\; \ne \; -->0,n\ge <!--\; \ge \; -->2,k\in <!--\; \in \; -->(0,n),a_k\in <!--\; \in \; -->\{0,1,2,\dots <!--\; \dots \; -->\}$, under what conditions is $f(r),f(r+1),\dots <!--\; \dots \; -->f(r+n)$ an arithmetic sequence for integer r?
When does f(x) not generate an $(n+1)$ term arithmetic sequence? (Update: Never - based on responses to this question).
Updated question: When does f(x) generate an n term arithmetic sequence? Given n values, we can fit an n-degree polynomial. The question is given the polynomial, when does it generate an n-term arithmetic sequence for n consecutive values of x
Note that $a_k$ are fixed in this problem.

ODE water drop modeling question
I have been working on a ODE homework which involves modeling the velocity of a drop of water falling from the sky. The ODE that models its velocity is given by:
$$m{v}^{\prime }=k{v}^2-<!--\; -\; -->mg,k=\frac{1}{2}{C}_d{\mathrm{\rho <!--\; \rho \; -->}}_a,$$
$C_d$: friction, ${\mathrm{\rho <!--\; \rho \; -->}}_a$: air density, A: transversal section of the water drop.
I have had to find the theoretical velocity limit of the water drop as a function of A by solving directly the ODE and compare these results with Euler and Runge-Kutta IV methods on MATLAB. I have done all of that.
The last question of my homework is an open question: It asks to modify the ODE presented by imagining now that the water drop losses a fraction of mass (by evaporation) while it is falling. I will have to apply Euler and Runge-Kutta IV on this new ODE. So I am looking for suggestions to improve the equation. Mass m now is going to be a function of time, it has to decrease. I have been thinking to assume that the water drops are spheres and by using $density\ast <!--\; \ast \; -->volume=mass$

Numbers raised to logarithmic expressions
I'm currently 11th grader in secondary school and we are learning logarithms. I think I know them quite well but I'm stuck on this exercise:
Evaluate $2^x-<!--\; -\; -->{10}^y$, where $x=1/lo{g}_6(2)$ and $y=2/lo{g}_2(10)$
When I plug the values in the calculator I get an answer of 10. But can someone show me the algorithm on how to calculate this? Because I'm unable to get the answer by myself (Sorry if some terms are not on point, I'm not a native English speaker.)

Can O(n + logn) be called O(n)?
I know that O(n + 5) would be classified as a worst-case runtime of O(n), but can I do that with logn too?

How to get $d$ in terms of $A$ and $B$
I'm trying to get $d$ in terms of $A$ and $B$ having the next equations:
$$0=A+B\ast <!--\; \ast \; -->{\log }_2<!--\; \; -->(d)$$
$$6=A+B\ast <!--\; \ast \; -->{\log }_2<!--\; \; -->(\frac{d}{2})$$
EDIT
How about A in terms of B and d? And B in terms of A and d?

Prove the logarithmic inequality
Prove that: $({\log }_{24}<!--\; \; -->48{)}^2+({\log }_{12}<!--\; \; -->54{)}^2>4$
I tried to put $t={\log }_2<!--\; \; -->3$ and get the equation
$6t^4+32t^3+22t^2-<!--\; -\; -->84t-<!--\; -\; -->74>0$. But I can't do anything with it...

Why is $$-<!--\; -\; -->\ln <!--\; \; -->x$$ is not equal to $$1/\ln <!--\; \; -->x$$?
I am doing differential equation now and I need to convert them into the proper form in order to do my homogeneous differential equation. So now I just found out that $$-<!--\; -\; -->\ln <!--\; \; -->x$$ is not equal to $$1/\ln <!--\; \; -->x$$. I thought it should be able to convert to $$-<!--\; -\; -->\ln <!--\; \; -->x$$ to the negative 1 then I can put it into the form $$1/\ln <!--\; \; -->x$$. Can anyone explain about it?

Simplify
$$4^{-<!--\; -\; -->n/2}\times <!--\; \times \; -->2^{(n+3)}\times <!--\; \times \; -->{16}^{-<!--\; -\; -->1/2}$$

How to get the results of this logarithmic equation?
How to solve this for $x$:
$${\log }_x<!--\; \; -->(x^3+1)\cdot <!--\; \cdot \; -->{\log }_{x+1}<!--\; \; -->(x)>2$$
I have tried to get the same exponent by getting the second multiplier to reciprocal and tried to simplify $(x^3+1)$