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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1
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1answer
47 views

Want to show this expression $>0$ (by picking constants carefully)

Suppose $A$, $B$, $C$ and $D$ are fixed arbitrary positive numbers. I am free to choose $\epsilon$, $\epsilon_1$ and $\epsilon_2$ but they must be positive. Can I choose the epsilons so that ...
7
votes
5answers
8k 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, ...
4
votes
4answers
115 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
1answer
278 views

Invariants in a second order equation

For $Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$, why are $\begin{vmatrix} A &B \\ B &C \end{vmatrix}$ and $\begin{vmatrix} A & B & D\\ B & C & E\\ D & E & F \end{vmatrix}$ ...
5
votes
3answers
354 views

What is the formula for this curve?

Three years of calculus in college have served me nothing, apparently, since I can't for the life of me remember even the basics. I'm working on a small software project where I have a table with say ...
4
votes
2answers
166 views

Supremum and greatest element

I have just learned what the definition is of a supremum, and I am confused to something my textbooks says: Subsets with a supremum don't have to have a greatest element, for example: $(0,3): = \{x ...
0
votes
1answer
101 views

Algebra question resubmited [closed]

Hello I have a quick algebra question. If I have the following expression $\large \frac{1}{x^2+1}$, and I multiply the numerator and the denominator by $(x^2+1)$. Is there any way I can get ...
1
vote
1answer
191 views

Simultaneous equations in four variables

I'm solving the following equations, $$x+y=zw$$ $$z+w=xy$$ How many solutions $(x,y,z,w)$ exist, if the variables are reals?
1
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2answers
887 views

How can I systematically find the roots of $ x^4 + 1?$ Is there some algorithm? [duplicate]

Possible Duplicate: How to find the root of $x^4 +1$ What algorithms can be used for finding all roots of the given polynomial: \begin{equation} x^4 + 1 = 0 \end{equation}
2
votes
3answers
69 views

Multiplication of a Range

Say I have a number, "a". 0 < a < 5. Let's say I have another number, "b". 0 < b < 10 Is ab < 50? If you were to write this as a * b < 5 * b, you would find that the upper bound ...
7
votes
4answers
365 views

Interesting GRE problem

I found this in a practice GRE problem. I thought I would have a crack at it after being spoiled by the answer At how many points in the xy-plane do the graphs of $y = x^{12}$ and $y = 2^x$ ...
3
votes
3answers
101 views

a typical polynomial root problem

I've been thinking and trying to solve this problem for quite sometime ( like a month or so ), but haven't achieved any success so far, so I finally decided to post it here. Here is my problem: ...
2
votes
4answers
2k views

Finding the cube root of a number using integer arithmetic only?

Is it possible to find cube root of a 150-200 digit decimal number correct truncated (not rounded ) upto 10 decimal places using integer arithmetic only? The question is an algorithmic one not a pure ...
6
votes
2answers
523 views

Solving a system of quadratic equations.

Solve for real $(a, b, c)$ satisfying $$ab + bc + ca = 1$$ $$a^2 − 2b^2 = 1$$ $$2b^2 − 3c^2 = 1$$ I try isolating $a$, but it leads to a very complicated expression in $a$.
3
votes
3answers
168 views

Simple triangle problem.

I am currently stuck on a problem that should be easy enough... In an isosceles triangle the two equal sides are $5m$ and the area is $12m^2$. How can I find out the length of the third side? My ...
2
votes
3answers
423 views

Proof for $\frac{x^n}{x^n}=x^{n-n}$

Does the equality $\forall n,x\not=0: \frac{x^n}{x^n}=x^{n-n}$ come straight from the definition of exponents, or is a more elaborate proof needed? Please note that I'm not asking about $x^0=1$, but ...
3
votes
3answers
161 views

Very simple triangle area problem

I have a problem with a triangle question. The question is: Imagine a triangle with the points $A(-5, 0), B(-3,-7), C(2,-2)$, where the sides $AB$ and $AC$ are equal. What is the area of this ...
2
votes
3answers
135 views

Finding the number of points on the straight line joining $(-4,11)$ and $(16,-1)$

Find the number of points on the straight line which joins $(-4,11)$ and $(16,-1)$ whose coordinates are positive integers. a) $1$ b) $2$ c) $3$ d) $4$
1
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3answers
414 views

Finding the sum of all solutions

$2x + 3y = n$ has exactly $2011$ non-negative integral solutions. Determine the SUM of the possible values of $n$.
3
votes
3answers
139 views

$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$

$$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$$ Is that correct? I would've done $$ 2^{10 - x} \cdot 2^{10 - x}\;\; = \;\; (2)^{10 - x + 10 - x} \; = \; (2)^{2 \cdot (10 - x)} \;=\; 4^{10 - x}\tag{1} $$ ...
1
vote
2answers
92 views

Show that if $\lfloor x+a \rfloor$ = $\lfloor x+b \rfloor, \forall x \in \Bbb R$ then $a=b$; is showing that $x+a=x+b$ enough?

If i show that $x+a=x+b$ only if $a=b$, does that prove that the above is also true? $ x+a=x+b \iff x+a-x-b=0 \iff a-b=0 \implies b=a$ also is this any good?
3
votes
4answers
363 views

Solving $|x^2 - 1| - 1 = 3x - 2$ without graphing

The book I'm working though has just showed how to find the intersection points of functions containing moduli by graphing them, visually spotting the intersection points, and then solving ...
2
votes
4answers
146 views

The product $\prod_{m=1}^{11} (x^m - m)$

What would be the co-efficient of $x^{60}$ in the expansion of $\space$ $\prod_{m=1}^{11} (x^m - m)$ ?
6
votes
3answers
182 views

Help with infinite sum

Can you guys give me a hint on evaluating $$\sum_{n=1}^\infty \frac{1}{n(n+2)(n+4)}?$$ I have tried partial fractions but the series is not telescopic (at least I cannot see it)...
3
votes
2answers
76 views

$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$

Given that the equation $$p(x)=a_0x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ has $n$ distinct positive roots, prove that $$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$$ I had ...
5
votes
3answers
301 views

Showing that complicated mixed polynomial is always positive

I want to show that $\left(132 q^3-175 q^4+73 q^5-\frac{39 q^6}{4}\right)+\left(-144 q^2+12 q^3+70 q^4-19 q^5\right) r+\left(80 q+200 q^2-243 q^3+100 q^4-\frac{31 q^5}{2}\right) r^2+\left(-208 q+116 ...
0
votes
1answer
50 views

How to derive the equation of a line going through a set of coordinates then give the next coordinate in the sequence?

How to derive the equation/function generating a set of coordinates and determine the next coordinate?
0
votes
1answer
165 views

How to show that for union of mutually disjoint, non empty subsets, distinct equivalence classes can be defined.

Prove that The distinct equivalence classes if an equivalence relation on $A$ provide us with a decomposition of $A$ as a union of mutually disjoint subsets. Conversely, given decomposition of $A$ ...
11
votes
7answers
518 views

Can someone show me why this factorization is true?

$$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \dots + xy^{n-2} + y^{n-1})$$ Can someone perhaps even use long division to show me how this factorization works? I honestly don't see anyway to "memorize ...
2
votes
1answer
263 views

Questions involving polynomials

I have been having problems with questions involving polynomials like asked in olympiads . A few examples of these type of problems I'm posting here: (please don't give me solutions to these ...
3
votes
1answer
126 views

Taking $e^{xn}=(e^x)^n$ a step further to rational numbers.

Working on an assignment but I've run into a stumbling block! I've got a couple of problems that I don't know how to do! The problems are attempting to have you define the $\log$ and $\exp$ ...
2
votes
1answer
44 views

Given the quadratic

Given the quadratic polynomial $ax^2 + bx + c$, find a new polynomial with coefficients expressed in terms $a$, $b$ and $c$ such that the product and the sum of its zeros will be the sum and the ...
0
votes
3answers
137 views

System of equations homework

Given the sytem of equation $y= -x+6$, $y= x/3+c$ with the solution lying in quadrant I, find all possible values of $c$.
1
vote
1answer
81 views

Find the sum of the first $50$ positive even integers

Mental math - The sum of the first $50$ positive odd integers is $50^2$. Find the sum of the first $50$ positive even integers.
0
votes
1answer
87 views

Question Regarding a Class of Students

A class has less than 30 students. Exactly 3/4 of them own footballs. Exactly 7/8 of them own football boots. i.How many students are there in the class? ii.How many students own both football ...
1
vote
2answers
57 views

Paper square diagonal

A paper square has diagonal length 6 in. two folders are made along lines perpendicular to the diagonal and through trisection points A and B. By what percentage has the visible area of the paper ...
2
votes
1answer
58 views

Rationalize Below Equation

How can I rationalize the following equation: $$\frac{22}{4\sqrt[3]{9}+2\sqrt[3]{6}+\sqrt[3]{4}}$$
4
votes
3answers
2k views

Writing Polar Equations In Parametric Form

For an example problem, in my textbook, the author wanted to demonstrate how to graph a polar function. Deeming it most convenient, my author took the polar function $r=2\cos 3\theta$, and re-wrote it ...
0
votes
1answer
31 views

Find the value of c for?

A value c for which the given equation has 2 distinct negative roots. $$ x^4 + 2cx^2 + 2cx + 1 + x^2 = 0$$ I solved it up to here, what does this imply ? $$ \sqrt{ 1+c^2 } > 2 - c $$
1
vote
1answer
45 views

Maximization of an integer input function

Maximize the value of the function $$ z=\frac{ab+c}{a+b+c}, $$ where $a,b,c$ are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, ...
20
votes
4answers
3k views

Purely “algebraic” proof of Young's Inequality

Young's inequality states that if $a, b \geq 0$, $p, q > 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$). Back when I ...
4
votes
1answer
82 views

How to find $1/x^3 + 1/y^3$?

If I am given, $x + y = a$ and $xy = b$, how would I find the value of $\dfrac1{x^3} + \dfrac1{y^3}$?
0
votes
1answer
34 views

Investments in M -HomeWork

Ryan invests M dollars in the stock market. After his investment increases by 10%, he takes out \$150 and sets it aside. The remaining investment then decreases by 10%. If the sum of the invested ...
1
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3answers
161 views

how to get from $\frac{x}{x+1}\;$ to $\;1 - \frac{1}{x+1}$

Please show me how to manipulate $\dfrac{x}{x+1}\;\;$ to get $\;\;1 - \dfrac{1}{x+1}$
1
vote
0answers
252 views

Precalculus word problem on rocket height

I recently wrote a test in my precalculus class, and came across a problem which I thought I did correctly, but everyone else taking the course seems to have gotten a different solution. Because of ...
4
votes
1answer
1k views

The trick to proving trigonometric identities

This question is motivated from the following excerpt from Rational Points on Elliptic Curves by Silverman and Tate: $$\cos \theta = \frac{1 - t^2}{1 + t^2}, \sin \theta = \frac{2t}{1+t^2}$$ ...
2
votes
2answers
181 views

Show that $f(x)$ has no repeated roots.

If $f(x)= \frac{x^n}{n!} +\frac{x^{(n-1)}}{(n−1)!}+···+x+1$,then show that $f(x)=0$ has no repeated roots.
1
vote
2answers
76 views

Closed Form for Simple Looking Sum?

Is there a closed form for the sum below? $$\sum_{s=0}^{m-1} \sum_{t=0}^{m-1} s~t~(m-s)~(m-t)\left|s-t\right|$$
1
vote
2answers
360 views

How to prove this simple statement: $\max\{a,b\}=\frac{1}{2}(a+b+|a-b|)$ [duplicate]

I am trying to prove this statement. for any $a,b \in \mathbb{R}$, $$\max\{a,b\}=\frac{1}{2}\big(a+b+|a-b|\big)$$ and $$\min\{a,b\}=\frac{1}{2}\big(a+b-|a-b|\big)$$ I am eating myself not knowing ...
0
votes
1answer
541 views

Write this piecewise function in terms of the unit step

\begin{align} f(t) = \begin{cases} 3t &\mbox{if } t \leq 3 \\ 12 & \mbox{if } 3<t\leq 7 \\ 0 & \mbox{if } t\geq 7 \end{cases} \end{align} I'm confused as to how I can write this in ...