# Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations and other symbolic-manipulation topics.

3,434 questions
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### Can you obtain $\pi$ using elements of $\mathbb{N}$, and finite number of basic arithmetic operations + exponentiation?

Is it possible to obtain $\pi$ from finite amount of operations $\{+,-,\cdot,\div,\wedge\}$ on $\mathbb{N}$ (or $\mathbb{Q}$, the answer will still be the same), note that set of all real numbers ...
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### Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and calculus....
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### Sum equals the product

A while ago, just playing with some numbers I noticed that $1+2+3=1\cdot2\cdot3$, so I started thinking about the non-zero integer solutions of the equation $$\prod_{i=1}^na_i=\sum_{i=1}^na_i$$ For ...
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### Solution by radicals of $(1+x)^n=x^m$

Given the equation $$(1+x)^n=x^m$$ where $m$ and $n$ are two different natural numbers, I was trying to find as many solution as possible expressing them without transcendent functions. WLOG we can ...
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### Does this “almost all integers in order” sequence have a closed form?

Can you help me define a formula for the following sequence (first $130$ terms) : ...
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### Product of squared sines: $\prod_{k=1}^{n-1}\prod_{j=1}^{n-1}\left[ \sin^2\left(\frac{k\pi}{2 n}\right) +\sin^2\left(\frac{j\pi}{2 n}\right)\right]$

I have a double product $$a(n)=2^{2 (n-1)^2} \prod_{k=1}^{n-1} \prod_{j=1}^{n-1} \left[ \sin^2 \left(\frac{k\pi}{2 n}\right) +\sin^2 \left(\frac{j\pi}{2 n}\right) \right]$$ which always gives ...
How many triples $(x,y) \in \mathbb{N^+}^2$ and $n \gt 1$ are there such that $x^n - y^n = 2^{100}$ I dont know how to start. Any hint will be helpful. I know the identity $x^n-y^n = (x-y)(... 0answers 76 views ### sum of the Series$\sum^{n}_{r=0}(-1)^r\binom{n}{r}\left[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+…\bf{m\; terms}\right]$The sum of the Series$\displaystyle \sum^{n}_{r=0}(-1)^r\binom{n}{r}\left[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+........\bf{m\; terms}\right]\bf{My\; Try::}$Let $$\displaystyle S= \... 0answers 174 views ### Proving that u_n is arithmetic sequence Let u_n be a sequence defined on natural numbers (the first term is u_0) and the terms are natural numbers (u_n\in \mathbb{N} ) We defined the following sequences:$$\displaystyle \large x_n=... 0answers 31 views ### Does there exist a function$f_{\Box,\Box}(\Box)$making the formula$a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$true? Let$a$and$b$denote the resistances of two resistors. If they're put in series, the total resistance is$a+b. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{... 0answers 118 views ### If f(x) is a polynomial function with integer coefficients find min value of f(12) f(x) is a polynomial function with integer coefficients satisfying f(1)=5 and f(2)=7. What is the smallest possible positive value of f(12)? I have no idea on how to begin with this question. ... 0answers 100 views ### Thinking like in ancient Greece The next result is attributed to Archimedes: The equation x^3-ax^2+(4/9)a^2b has a positive root if and only if a>3b, where a and b are positive real numbers. This problem appeared in the ... 0answers 150 views ### Show \lim_{n\to \infty}{|A_{n}|+|A_{n-2}| \above 1.5pt |A_{n-1}|}=2 Question: Let A(n) be a finite square n \times n matrix with entries a_{ij}=1 if i+j is a prime; otherwise equals to 0. I write |A(n)| to count the number of 1's in A(n). Is it ... 0answers 212 views ### Is it possible to simplify a nested radical in the form \sqrt[3]{\sqrt[3]{A}-B} into \sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c} I'm wondering if there is a way to simplify any nested radical in the form \sqrt[3]{\sqrt[3]{A}-B} into \sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}. Examples such as \sqrt[3]{7\sqrt[3]{20}-1}=\sqrt[3]{\... 0answers 118 views ### Probability of another 3 integers with same sum and product as the first 3 integers Let us suppose 3 integers are selected at random from a large range, say$$-1000\leq x\leq y\leq z\leq 1000$$Now, we define the sum and product:$$\begin{align*}s&=x+y+z \\p&=xyz\end{align*... 0answers 99 views ### Reference? filler: IRS, Rhind Papyrus, High-school algebra I believe something like this was included as a filler in one of the MAA journals many years ago. I am searching for the exact reference (for the filler, or an earlier source). Someone dies, and ... 0answers 58 views ###a$and$b$are solutions of$ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $,$a+b=?a$and$b$are solutions of $$\frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0$$ What is$a+b=?$ Are there better approaches than the one below? Solution: ... 0answers 51 views ### Interesting applications of density to prove difficult theorems If we wanted to prove certain statements for every element of a set$S$, a possible approach is to prove the statement for a certain dense subset$S'\subset S$(with respect to a certain metric), then ... 0answers 80 views ### What is rule for when solving algebraic equations? I'm a high school student trying to get critical intuition when learning algebraic equation solving. For$x$any complex number and$c$constant, simple polynomial such as$x^n -c=0$are easily ... 0answers 122 views ### Prove that if$a,b,c \in \mathbb{R^+}\text{ and } abc=8\text{ then } {ab+4\over a+2}+{bc+4\over b+2}+{ca+4\over c+2}\ge6$Question: Prove that if$a,b,c \in \mathbb{R^+}\text{ and } abc=8\text{ then } {ab+4\over a+2}+{bc+4\over b+2}+{ca+4\over c+2}\ge6$My Approach: Now we have: $${ab+4\over a+2}={2\times (ab+4)\... 0answers 71 views ### Given n integers, is it always possible to choose m from them so that their sum is a multiple of m? The original question: given 6666 integers, (positive, negative or 0) is it always possible to choose 2018 from them so that the chosen numbers add to a multiple of 2018? (positive multiple, ... 0answers 111 views ### How to find self intersection of this polar curve: r = 1 + 2\cos(\theta) - 4\cos^2 (\theta) I am trying to find the values of \theta where the following polar curve intersect itself:$$r = 1 + 2\cos(\theta) - 4\cos^2 (\theta).$$I am studying this function on [0, 2\pi] since it is ... 0answers 88 views ### Real root of x(x+t)^2-4=0 \ge greatest real root of 4x^6-6x^4+4t^3x^3+t^6-3t^4=0 I have a problem that I can not solve although I know that the statement is true. For all t\ge0 the real root of x(x+t)^2-4=0 is greater than or equal to the largest real root of 4x^6-6x^4+4t^... 0answers 68 views ### Does (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 hold for n \leq 8? Let positive reals \{a_i\}, where not all a_i are equal. Does$$ f(\{a_i\}) = (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 $$hold for n \le 8? It is ... 0answers 119 views ### On Pandigital numbers The other day I ran across a very nice video from Numberphile in YouTube, which proves that the following formula is approximate to e by 18457734525360901453873570 digits, which is pretty amazing. ... 0answers 118 views ### If 2f(\frac{x}{2})+3f(\frac{2-x}{3})=g(x) then find 25ad-bc 2f(\frac{x}{2})+3f(\frac{2-x}{3})=g(x), 0 \leq x < 3, f''(x)>0; if g(x) is strictly increasing in (a,b) and g(x) is strictly decreasing (c,d) then 25ad-bc is ? My Attempt: ... 0answers 81 views ### Find the least number N such that N=a^{a+2b} = b^{b+2a}, a \neq b. When I graphed the relation a^{a+2b}=b^{b+2a} , it gives a graph similar to y=x. However, the question explicitly states that a \neq b. So does that mean that no such N exists ? What happens ... 0answers 77 views ### Equality of Floors of some Partial Sums Let S_n=\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!} denote the nth partial sum in the series expansion for e=\sum\frac{1}{k!}. I want to prove that \lfloor n\cdot(S_n+1/n!)\rfloor=\lfloor n\... 0answers 85 views ### Is this a valid way for performing polynomial division? While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ... 0answers 84 views ### If f(n)= \binom{n}{0}a^{n-1}-..+(-1)^{n-1}\binom{n}{n-1}a^{0} ,Then f(2007)+f(2008) If \displaystyle a= \frac{1}{3^{223}}+1 and \displaystyle f(n)= \binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+...........+(-1)^{n-1}\binom{n}{n-1}a^{0} Then value of f(2007)+f(2008) = \bf{My\; ... 0answers 165 views ### Finding the minimum value of a radical expression If a, b and c are positive real numbers, find the minimum value of \sqrt { \frac { a }{ b+c } } +\sqrt [ 3 ]{ \frac { b }{ c+a } } +\sqrt [ 4 ]{ \frac { c }{ a+b } } . I am not able to ... 0answers 295 views ### Motivation of Vieta's transformation The depressed cubic equation y^3 +py + q = 0 can be solved with Vieta's transformation (or Vieta's substitution) y = z - \frac{p}{3 \cdot z}. This reduces the cubic equation to a quadratic ... 0answers 150 views ### How to solve this equation in \mathbb{C} ? From a small simple calculation , we get the following formulas: \begin{cases} e^x = \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \Big) + \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n+... 0answers 244 views ### Writing sum of square roots with symmetric polynomials I want to write the function$$ F_N=\sum_{i=1}^N\sqrt{x_i} $$in terms of the N elementary symmetric polynomials of the N positive variables x_1,\dots,x_N. The N=1 case is trivial, as we ... 0answers 94 views ### Irreducibility of some polynomial Let p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1, for some k \geq 2 fixed. I would like to know if p(x) is irreducible in \mathbb{Q}[x]. 0answers 121 views ### Is there a name for systems of equations with min and max functions included? In a big project I'm working on, I'm running into systems of equations that look like the following:$$a = \min(b, c)b = d^2 + ac = \max(a + b, d)$$Basically, nonlinear systems of ... 0answers 58 views ### Existence of Solutions of Two Cubic Equations in a Particular Region If I have two cubic equations in two variables, ax^3 + bx^2 y + cxy^2+\dots=0 and another one with different coefficients, and I know that (x,y)=(0,0) or (1,1) are solutions, are there any nice, ... 0answers 57 views ### Need f:[0,\infty)\to[0,\infty) such that f is not convex but f(x^p) is for p>1. To be more specific I need to find an f:[0,+\infty)\to[0,+\infty) which satisfies the following: (somewhat trivial stuff) The function f is continuous, nondecreasing, there exists k>0 such ... 0answers 147 views ### Polynomial bound Let P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0 such that$$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$Suppose that P(x)> 0 for all ... 0answers 97 views ### Digits of two irrational numbers, given their power with fixed number of digits I have a, b \in \mathbb{R} \setminus \mathbb{Q}, I want to know the result of a^b, but I don't know exact a, b because I write them in numeric form. My question is how many digits of a, b have ... 0answers 424 views ### Mandelbrot set's border in parametric form I've post this question just because I'm curious, Mandelbrot set is defined as: z_{n+1} = z^2_n + c , if n \rightarrow \infty and it doesn't diverge we get the border. This border is unlimited ... 0answers 192 views ### How to transform series of series into series I need to prove this equation.$$ \sum_{k=0}^{i-2} \left( e \space α(k+1)\space\frac{(-1)^{i+k+2}}{(i-k-2)!} \right) = \sum_{k=0}^{i-2} \frac{(i-k)^k}{k!} \space e^{i-k} (-1)^k\space where,\space α(i) ... 0answers 176 views ### Functional inverse of$(a + b\sin\theta)^2\tan\theta$So, I revisited to the situation involved in my previous question, with the intent of generalizing it to any two masses and charges. When I started going through the model again, beginning with the ... 0answers 246 views ### Is it possible to express the product of two real numbers in terms of their difference? Some motivation: I have a (simplified) function$f(x) = \frac{k}x$, where$k$is some real constant and$x$is a real number. The result of the function, however, is always truncated to an integer for ... 0answers 50 views ### Problem about linear combinations of real numbers over$\Bbb Q$I have the following problem that might be silly but I am not able to find a solution. Let$a_1,\dots,a_n$be real numbers such that$a_1,\dots,a_n,\pi$are linearly independent over$\Bbb Q$. Let$...
Consider a function : $$f(x)= P(x)e^{-(x^4+2x^2)}$$in the domain $x \in (-\infty,\infty)$, $P(x)$ is any polynomial of degree $k$. What is the maximum possible number of extrema of the function. ...