Expert Answers to Algebra 2 Problems

$\underset{x\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}(\frac{\ln <!--\; \; -->(x-<!--\; -\; -->2)}{\ln <!--\; \; -->(x-<!--\; -\; -->1)}{)}^{x\ln <!--\; \; -->x}.$
L'Hopital seems like a very hardcore solutions given the situation.Are the any other options?

If a passenger train travels three times as fast as a freight train, and the freight train takes 4 hours longer to travel 210 miles, what is the speed of each train?

An engineer traveled 170 mi by car and then an additional 750 mi by plane. The rate of the plane was three times the rate of the car. The total trip took 7 h. Find the rate of the car.

Finding an expression for the sum of n tems of the series $1^2+2^2+3^2+...+n^2$
I know that if you have a non-arithmetic or geometric progression, you can find a sum S of a series with the formula $S=f(n+1)-<!--\; -\; -->f(1)$ where the term $u_n$ is $u_n=f(n+1)-<!--\; -\; -->f(n)$. Then you can prove that with induction.
What I don't understand is how I should go about finding the function f(n). For example if I want to calculate the sum to n terms of the series $1^2+2^2+3^2+...+n^2$ then, according to my textbook, my f(n) function should be a polynomial with degree one more than the degree of a term in my sequence - so because the nth term in the sequence is $P(n)=n^2$ then the function f(n) should be $f(n)=a{n}^3+b{n}^2+cn+d$. But how did they know that it should look like that and how do I gain some intuition into finding that function to help me solve similar problems in the future?

Numbers of solutions of the equation ${\log }_3<!--\; \; -->\frac{2x^2+3x+3}{5}=\frac{1}{{\log }_{2x^2+3x+9}<!--\; \; -->9}$
Pretty straightforward question. When I solved it, I got two positive and two negative solutions, so that would make 4 in total. None get discarded as the arguments in the logarithm still stay positive.
However, the solution is supposed to be 2, so I'm not sure where I went wrong.
I got the values of $x$ as $-<!--\; -\; -->2$, $1/2$, $12/5$ and $-<!--\; -\; -->18/5$

Galois theory in a different setting
Suppose that instead of wanting to express the roots of a polynomial equation with arithmetic operations and radicals we instead wanted expressed it with arithmetic operations and sin(x) ?
What (if anything) does Galois theory say about this situation?

Convolution product of arithmetic functions
An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions f and g to be the arithmetic function $f\star <!--\; \star \; -->g$, where
$(f\star <!--\; \star \; -->g)(n)=\underset{ij=n}{\sum <!--\; \sum \; -->}f(i)g(j),\text{ and }{f}^{\star <!--\; \star \; -->k}=f\star <!--\; \star \; -->f\star <!--\; \star \; -->\cdots <!--\; \cdots \; -->\star <!--\; \star \; -->f(k\text{ times}).$
We say that two arithmetic functions f and g are dependent if there exists a nontrivial polynomial of two variables $P(x,y)=\underset{i,j}{\sum <!--\; \sum \; -->}{a}_{ij}{x}^i{y}^j$ with real coefficients such that
$P(f,g)=\underset{i,j}{\sum <!--\; \sum \; -->}{a}_{ij}{f}^{\star <!--\; \star \; -->i}\star <!--\; \star \; -->g^{\star <!--\; \star \; -->j}=0$
and say that they are independent if they are not dependent. Let p and q be two distinct primes and set
$f_1(n)=\{\begin{array}{l}1\text{if }n=p,\\ 0\text{otherwise};\end{array}{f}_2(n)=\{\begin{array}{l}1\text{if }n=q,\\ 0\text{otherwise}.\end{array}$
Prove that $f_1$ and $f_2$ are independent.
My book says that
$f_1^{\star <!--\; \star \; -->i}\star <!--\; \star \; -->f_2^{\star <!--\; \star \; -->j}(n)=\{\begin{array}{l}1\text{if}n=p^i{q}^j\\ 0\text{otherwise}\end{array}$
and the result follows from that, but how do they get that and how does the result follow from that?

How many distinct degree 7 polynomials are there over the modular arithmeic modulo 7?
If it's infinite, is it countable or uncountable infinite?
I am a newbie to this topic... I don't know what modular arithmetic for polynomials means. Can someone please give me a link where I can learn?

A rancher has 4300 ft of fencing available to enclose a rectangular area bordering a river. He wants to seperate his cows and horses by dividing the enclosure into 2 equal areas. If no fencing is required along the river, find the length of the center portion that will yield that maximum area.

A definite integral contianing ln(x)
everyone, I met a tough definite integral as follows,
$I=\underset{1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\int <!--\; \int \; -->}}\frac{\ln <!--\; \; -->x}{{\left(x+a\right)}^m{\left(x+b\right)}^{n+1}}dx,$
where a and b are constant, m and n are non-negative integers.

Two Big Macs and three Whoppers contain 520 mgs of cholesterol. Three Big Macs and one whopper contain 353 mgs of cholesterol. How many mgs of cholesterol are in each burger?

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-<!--\; -\; -->x^2}{12}){]}^{\frac{3}{2}}$?
For the given function, the question is whether, f(x) can equal 1 for some real value of x?

Integral roots for $f(x)=41$ if $f(x)=37$ has 5 distinct integral roots.
Given a polynomial f(x) with integral coefficients and $f(x)=37$ has 5 distinct integral roots, find the number of integral roots of $f(x)=41$?
My Approach: Say $f(x)=(x-<!--\; -\; -->r_1)(x-<!--\; -\; -->r_2)(x-<!--\; -\; -->r_3)(x-<!--\; -\; -->r_4)(x-<!--\; -\; -->r_5)g(x)+37$, where ri are the distinct integers.
Now for $f(x)=41$ we have $(x-<!--\; -\; -->r_1)(x-<!--\; -\; -->r_2)(x-<!--\; -\; -->r_3)(x-<!--\; -\; -->r_4)(x-<!--\; -\; -->r_5)g(x)=4$, so the factors can be $\pm <!--\; \pm \; -->1,\pm <!--\; \pm \; -->2$ or $\pm <!--\; \pm \; -->4$. Given $r_i$ are distinct at most two of them will give $\pm <!--\; \pm \; -->1$, then there can be both of $\pm <!--\; \pm \; -->2$ or one of $\pm <!--\; \pm \; -->4$. This is where I get lost, since even if I use all of $\pm <!--\; \pm \; -->1,\pm <!--\; \pm \; -->2$, I will be still be left with one $x-<!--\; -\; -->r_i$ factor. What about that? Does it matter that 37 and 41 are primes, or is it just a coincidence?

Explain what is wrong with the “obvious” approach to polynomial division using a point-value representation, i.e., dividing the corresponding y values. Discuss separately the case in which the division comes out exactly and the case in which it doesn’t.

Prove that ${\log }_a<!--\; \; -->(1/x)=-<!--\; -\; -->{\log }_a<!--\; \; -->(x)$
But it has two problems: when x=0 and on the other problem it doesn't mention any condition. How should I solve it in each of them?

You invest $50,000, part at 3% simple interest and part at 4% simple interest for one year. How much money did you invest in each account if the interest earned in the 3% account was $30 greater than the interest earned in the 4% account?

Is $1+\lg <!--\; \; -->i=\lg <!--\; \; -->(i+i)$?
I've been studying Sedgewick's "Algorithms" book and in proof of one proposition he writes the following:
the property is preserved because
$1+\lg <!--\; \; -->i=\lg <!--\; \; -->(i+i)\le <!--\; \le \; -->\lg <!--\; \; -->(i+j)=\lg <!--\; \; -->k$
I cannot wrap my brain around the first part of this inequation, namely $1+\lg <!--\; \; -->i=\lg <!--\; \; -->(i+i)$. Can anyone offer an explanation? Thanks in advance!

Find the leftmost (most significant digits) of a large exponent calculation, say ${99}^{99}$
I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, ${99}^{99}=3.697296\times <!--\; \times \; -->{10}^{197}$
I only need to know the digits $3697296$
Is there any way of finding this without doing the full calculation?

Logarithm rule for branch cut logarithms
I know that for $a,b\in <!--\; \in \; -->\mathbb{R}$, the rule $\log <!--\; \; -->(ab)=\log <!--\; \; -->(a)+\log <!--\; \; -->(b)$
What about for $a_1,b_1$ in the right half-plane, or $a_2,b_2$ in the sector from $\frac{-<!--\; -\; -->3\mathrm{\pi <!--\; \pi \; -->}}{4}$ to $\frac{3\mathrm{\pi <!--\; \pi \; -->}}{4}$? For the logarithm, take a branch cut on the negative real line.

Describe the sampling distribution of
ModifyingAbove p with caretp.
Assume the size of the population is 20,000.
$$\begin{gathered} n=700 \\ p=0.4420 \end{gathered}$$