Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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7
votes
7answers
531 views

What is the result of $\lim\limits_{x \to 0}(1/x - 1/\sin x)$?

Find the limit: $$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)$$ I am not able to find it because I don't know how to prove or disprove $0$ is the answer.
3
votes
4answers
1k views

Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$

I want to solve this system of equations, I have been out of Maths for a long!! $$\sqrt{x}+y=7$$ $$\sqrt{y}+x=11$$ Just wondering easiest step to find values for $x$ and $y$ from the above ...
1
vote
4answers
1k views

How to factor the quadratic polynomial $2x^2-5xy-y^2$?

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
26
votes
11answers
4k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
9
votes
7answers
600 views

Derivation of the general forms of partial fractions

I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take (1) If the denominator of the fraction has linear factors: $${5 \over {(x - 2)(x + 3)}} \equiv {A ...
5
votes
3answers
356 views

$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-…}}}} $, what is the nearest value of $y^2 - y$?

I found this question somewhere and have been unable to solve it. It is a modification of a very common algebra question. $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $, what is the ...
7
votes
4answers
566 views

How does partial fraction decomposition avoid division by zero?

This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example: $$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$ Multiplying ...
7
votes
4answers
552 views

How to detect when continued fractions period terminates

I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates? Let me give you an example: $\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
8
votes
2answers
812 views

Finding roots of polynomials with rational coefficients

I'm looking for a general approach (or approaches) for finding the roots of polynomials with rational coefficients of higher degrees than $4$. The problem is that I need to find the exact roots and ...
4
votes
4answers
196 views

Quadratic substitution question

By using the substitution $p=x+\frac{1}{x}$, show that the equation $$2x^4+x^3-6x^2+x+2=0$$ reduces to $2p^2+p-10=0$. I can't think of anything that produces a useful result, I tried writing p as ...
4
votes
2answers
504 views

How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?

This sum is difficult. How can I compute it, without using calculus? $$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$ If someone can explain some technique to do it, I'd appreciate it. Or advice using ...
0
votes
3answers
109 views

Finding the range of $f(x) = 1/((x-1)(x-2))$

I want to find the range of the following function $$f(x) = \frac{1}{(x-1)(x-2)} $$ Is there any way to find the range of the above function? I have found one idea. But that is too critical. Please ...
41
votes
10answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
35
votes
6answers
1k views

Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
21
votes
3answers
591 views

Why everytime the final number comes the same?

I have come across an interesting puzzle. Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with $\text{Number}_{new} = xy/(x + y)$ OR $\text{Number}_{new}= x + y + xy$ ...
6
votes
2answers
900 views

Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$

Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that: $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} ...
14
votes
4answers
496 views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
11
votes
5answers
3k views

How can I write an equation that matches any sequence?

One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is: x 0 1 2 y 1 5 9 If ...
8
votes
6answers
576 views

Is it wrong to say $ \sqrt{x} \times \sqrt{x} =\pm x,\forall x \in \mathbb{R}$?

Is it wrong to say $$ \sqrt{x} \times \sqrt{x} =\sqrt{x^2}= \pm x$$ I am quite sure that $\sqrt{(x)^2} = \pm(x)$ But, does $\sqrt{x } \times \sqrt{x} =- (x)$ doesn't holds in $\mathbb{R}$ but if we ...
2
votes
3answers
1k views

Create polynomial coefficients from its roots

Given some roots : $r_1,r_2,\ldots,r_n$, how can we reconstruct polynomial coefficients? I know the Horner scheme and that we can just go backwards receiving those coefficients. But I'm curious if ...
14
votes
7answers
653 views

$n^5-n$ is divisible by $10$?

I was trying to prove this, and I realized that this is essentially a statement that $n^5$ has the same last digit as $n$, and to prove this it is sufficient to calculate $n^5$ for $0-9$ and see that ...
8
votes
1answer
265 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
7
votes
12answers
10k views

Derivation of the formula for the vertex of a Parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $y = a x^2 + b x + c$ My teacher gave me the formula: $x = -\frac{b}{2a}$ as the $x$ ...
5
votes
1answer
833 views

Why not write the solutions of a cubic this way?

For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as: $$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + ...
3
votes
2answers
112 views

find extreme values of $\frac{2x}{x²+4}$

I am doing my homework like a good little boy. I know that when I want to find the extreme values of a function I have to put the derivative equal to zero so I can find the x values. I've done it ...
3
votes
4answers
509 views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
2
votes
4answers
598 views

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ show that $x=-c/b$ when $a=0$

OK, this one has me stumped. Given that the solution for $ax^2+bx+c =0$ $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\qquad(*)$$ How would you show using $(*)$ that $x=-c/b$ when $a=0$ (Please dont use $a=0$ ...
1
vote
1answer
193 views

Establishing formula from recurrence

Can anyone tell me how do we establish a formula from a given recurrence relation? Take the example of $f(n) = 2f(n-1) + 1$, $n \in \mathbb{Z^+}$, $f(1) = 1$ When I write out the first few values, it ...
0
votes
3answers
167 views

Find the equation of the line form

Find the equation of the line that goes through the points $(2, 9)$ and $(8, 6)$. Write the equation in standard form.
9
votes
5answers
599 views

Algorithms for “solving” $\sqrt{2}$

The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
6
votes
3answers
663 views

Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
5
votes
2answers
424 views

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have ...
5
votes
4answers
1k views

Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
4
votes
1answer
2k views

Number of surjective functions from A to B

Am I on the right track? I am not sure about my reasoning... Number of surjective functions from $A$ to $B$ $$A = \{1,2,3,4\} ; B = \{a,b,c\}$$ We must count the surjective functions, meaning the ...
4
votes
4answers
178 views

Not so easy optimization of variables?

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to $2x^2+5xy+3y^2=2$ and $6x^2+8xy+4y^2=3$. (calculus is not allowed). I tried everything I could but whenever I got for example ...
4
votes
4answers
110 views

Finding range of $\frac{x}{(1-x)^2}$

I'm working though some exercises. One of them is asking to find the range of $f(x) = \frac{x}{(1-x)^2}$. The chapter this exercise belongs to is after the one where the differentiation power rule is ...
4
votes
3answers
318 views

How to prove floor identities?

I'm trying to prove rigorously the following: $\lfloor x/a/b \rfloor$ = $\lfloor \lfloor x/a \rfloor /b \rfloor$ for $a,b>1$ So far I haven't gotten far. It's enough to prove this instead ...
4
votes
5answers
381 views

Showing $a^2 < b^2$, if $0 < a < b$

Lately, I've been stumbling with proofs of inequalities. For example: Given $0 < a < b$ Show $a^2 < b^2$ The only thing I've been able to come up with so far: $a^2 < b^2$ ...
3
votes
4answers
1k views

Find an equation of the plane

I have come across a difficult problem in my textbook. The problem ask: Find an equation of the plane that passes through the line of intersection of the planes $x-z=1$ and $y+2z=3$ and is ...
2
votes
4answers
523 views

For what natural numbers is $n^3 < 2^n$? Prove by induction

Problem For what natural numbers is $n^3 < 2^n$? Attempt @ Solution For $n=1$, $1 < 2$ Suppose $n^3 < 2^n$ for some $n = k \ge 1$ It looks like the inequality is true for $n = 0$, $n = 1$ ...
0
votes
5answers
379 views

How could we solve $x$, in $|x+1|-|1-x|=2$?

How could we solve $x$, in $|x+1|-|1-x|=2$? Please suggest a analytical way that I could use in other problems too like this $ |x+1|+|1-x|=2$ and of this genre. Thank you,
-4
votes
2answers
292 views

Integral eigenvectors and eigenvalues

I need to find the eigenvalues e eigenvectors of this integral. a) $$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$ b)- Solved thanks $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
45
votes
12answers
3k views

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
25
votes
2answers
1k views

Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?

I'm currently studying exponential functions, however I've come across this before and it was just a law that I was told to accept without much proof or explanation, could someone explain the ...
56
votes
15answers
4k views

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
11
votes
7answers
1k views

Factorize the polynomial $x^3+y^3+z^3-3xyz$

I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
6
votes
5answers
5k views

what is the best book for Pre-Calculus?

i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, ...
9
votes
2answers
610 views

Proof of an inequality: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$ [duplicate]

Possible Duplicate: Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction How do I prove the following? $$\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + ...
8
votes
3answers
241 views

A trigonometric equation

Some years ago, I came across the following question: Find the value of $\tan5°\tan55°\tan65°\tan75°$. The value is 1 and I generalized the question as follows. $\tan a°\tan b°\tan c°\tan d° ...
26
votes
5answers
1k views