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.

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Does the remainder theorem work for constant function?

The remainder theorem stated that " If a polynominal f(x) is divided by (x-k), the remainder is f(k)." So I think of a constant function ,say, f(x)= 5 and I divide this by (x-1). According ...
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What is the difference between the principal square root of $x$ and $x^{1/2}$?

Today I learned that the square root symbol ($\sqrt{}$) represents only the "principal" square root of a number. What about exponents?—if I write $x^{1/2}$, would that encompass the negative ...
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2 votes
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63 views

Proof from Lang’s Basic Mathematics

Starting on page 11, I don’t understand the proof. If $a$, $b$ are negative integers, then $a + b$ is negative. Proof. We can write $a = -n$ and $b = -m$, where $m, n$ are positive. Therefore $a + b = ...
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2 answers
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Simplify $x/(x-2)+(x-1)/(x+1)=-1$ step by step

Simplify $$x/(x-2)+(x-1)/(x+1)=-1$$ step by step. So I clear the fractions by multiplying by the common denominator of $(x-2)(x+1)$ and you have $$x(x+1) + (x-1)(x-2) = -(x-2)(x+1)$$ $$x^2 + x + x^2 - ...
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2 votes
2 answers
131 views

Why $f^2(x) \ne f(x)^2$?

I am working on an exploration which starts with the following affirmation: In this section you studied the Binomial theorem. Recall function composition from earlier in the course. In this context (...
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2 answers
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Is $ \int _{1/a}^af'\left(x\right)×f\left(x\right)\:dx = 0$ true given $f\left(x\right) = bx + \frac{b}{x^n}$?

In my math final exam earlier this week there was a function: $$f\left(x\right) = 3x + \frac3x$$ And its derivative, $$f'\left(x\right) = 3 - \frac{3}{x^2}$$ And one of the questions about this ...
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0 answers
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Doubt from permutation

Let $Mn=(0.a1.a2....an-1.an)$ be set of fractions such that ai=0 or 1 for i belongs to (1,2,....n-1) and an=1. If Tn and Sn be number and sum of all elements inn Mn then find lim $n \to _\infty$ Tn/Sn ...
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1 answer
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How do you multiply $x < y$ to get $1/x > 1/y$?

I'm stuck on this part. I only know that I'd get $x/x = 1$ and $y/y = 1$, which doesn't fit the inequality. Rudin, page 8, Proposition 1.18e
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Question about differential

I am trying to do the following question: Determine whether each of the following line integrals is independent of path. If it is, find a function $h$ such that $d h=P d x+Q d y$. If it is not, find a ...
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Solve the expression [closed]

Prove that the value of the expression does not depend on a, b, c and x enter image description here I have gotten up to this point: enter image description here The solution in the book is 4, but I'm ...
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-3 votes
0 answers
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Why I need to do exponents before addition. [closed]

Here why addition is done first? We should do exponent according to pedmas rule example $(2+3)^2=5^2=25$
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2 votes
0 answers
29 views

Generalizing Ramanujan cubic denesting formula to higher powers

We have the following theorems for denesting radicals of degree 2 and 3 : Denesting theorem for degree 2 : If $\alpha, \beta$ are the roots of the equation, \begin{equation} x^2-ax+b = 0 \end{equation}...
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Understanding the difference between a divisor and a factor

Consider an arbitrary polynomial of degree four $$ P(x)=ax^4+bx^3+cx^2+dx+e,$$ where $a,b,c,d,e\in\mathbb{R}$. If $P(x)$ is divisible by a quadratic, say $(x-1)(x+9)$, are $x-1$ and $x+9$ factors of $...
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Mary (M) is twice as old as Ann (A) was when M was half as old as A will be when A is $3$ times as old as M was when M was $3$ times as old as A was.

The combined ages of Mary and Ann is $44$ years, and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as ...
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2 answers
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$3(2x+d)+c(x+5)=10x+17$

Given "$3(2x+d)+c(x+5)=10x+17$" what are the values of c and d. Upon expansion we get $6x+3d+cx+5c=10x+17$, meaning c must be equal to 4. I was playing around with the equation and I found ...
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1 answer
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Finding the equation of a parabola from its graph [closed]

can chat on discord but need help asap really struguling in this class
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6 votes
3 answers
74 views

Remarquable identities $f(n) = \frac{a^n}{(a-b)(a-c)} + \frac{b^n}{(b-a)(b-c)} + \frac{c^n}{(c-a)(c-b)}$

Let $n$ be an integer, and \begin{equation} f(n) = \frac{a^n}{(a-b)(a-c)} + \frac{b^n}{(b-a)(b-c)} + \frac{c^n}{(c-a)(c-b)} \end{equation} \begin{equation} g(n) = \frac{(bc)^n}{(a-b)(a-c)} + \frac{(ac)...
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1 vote
1 answer
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Why is $\lim_{x\to1}f(x)g(x+1)$ undefined, where $f(x)=2$ if $x=1$, $f(x)=\cos(\pi x/2)$ if $x\neq1$, $g(x)=1$ if $x<2$, and $g(x)=-1$ if $x\geq2$?

Consider $$f(x)=\begin{cases}2&\text{if }x=1\\\cos\left(\frac{\pi x}2\right)&\text{otherwise}\end{cases}$$ and $$g(x)=\begin{cases}\phantom{-}1&\text{if }x<2\\-1&\text{otherwise}\...
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A positive integer gets reduced by nine times when one of its digits is deleted and the resultant number is divisible by 9.

The question says A positive integer gets reduced by nine times when one of its digits is deleted and the resultant number is divisible by 9. Prove that to divide the resultant number by 9, it is ...
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Prove that $\lim_\limits{n \to \infty} \left(1+ \frac{x}{n}\right)^n =\lim_\limits{n \to \infty} \left(1+ \frac{1}{n}\right)^{nx}$ [duplicate]

We know that $\lim_\limits{n \to \infty} \left(1+ \frac{x}{n}\right)^n = e^x$, and this implies that $e = \lim_\limits{n \to \infty} \left(1+ \frac{1}{n}\right)^{n} $ However, if you raise both sides ...
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2 answers
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How to use a function in the middle of an equation

I'm reading up on some electrical engineering and have been presented with an expression which I'm unfamiliar with. I've taken College Algebra and I am familiar with the idea of a function in the ...
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1 vote
1 answer
34 views

Is $-\sin(t)\cos(t)$ a parabolic function of $-\sin(t)+\cos(t)+1$?

Suppose we have the following functions with shared parameter $t$: $$x(t) = -\sin(t)+\cos(t)+1$$ $$y(t) = -\sin(t)\cos(t)$$ When we plot them together as a planar curve we can see what appears to be ...
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1 vote
1 answer
119 views

If $f(x)=\sqrt{1+\sqrt{x+\sqrt{x^2 +\sqrt{x^3 +\cdots}}}},$ then find the value of $f(4).$ [duplicate]

If $f(x)=\sqrt{1+\sqrt{x+\sqrt{x^2 +\sqrt{x^3 +\cdots}}}},$ then find the value of $f(4).$ My attempt: $f(x) = \sqrt{1+{\sqrt{x}} f(x)} \implies f(4) =\sqrt{1+2f(4)} \implies f(4)(f(4)-2)=1.$ I don't ...
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Does there exist a numerical base such that $9 + 10 = 21$ is true?

We can set up the equation as follows: $$9 \text{ (base b)} + 10 \text{ (base b)} =21 \text{ (base b)} $$ And rewrite as $9_b +10_b = 21_b$ for a more compact notation. I can see that the base, $b$, ...
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1 answer
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Does there exist $a,b,c\in \mathbb{R}$ s.t. $\cos(t)\sin(t)t = a \exp \left\{ \left( \frac{\cos(t)+\sin(t)+t-b}{c} \right)^2 \right\}$?

Suppose that we have two functions of a shared parameter $t$: $$x(t) = \cos(t)+\sin(t)+t$$ $$y(t) = \cos(t)\sin(t)t$$ When I plot $x(t)$ against $y(t)$ I get the sense that the graph might be 'half-of-...
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9 votes
1 answer
131 views

Easier way to solve equation systems of $a+b+c+\cdots{}= 1$, $a^2 + b^2 + c^2+\cdots{}=2$ and so on without having to crunch massive expressions

I study at below college level. I have been trying to solve certain systems of equations involving $n$ equations of $n$ unknowns. For example, for $2$ unknowns, the problem is \begin{align} a^{\...
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-4 votes
0 answers
61 views

Proof by contradiction theory number [closed]

I try to prove this proposition by contradiction Show that if $n=kp$ and $k$ $>$ $ \sqrt{n} $ where $k$ and $p$ are positive integers then $ p \leq \sqrt{n} $ Suppose that $p,n \in \mathbb Z^+ $ ...
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-2 votes
0 answers
31 views

quadratic function cutoff analysis,some ideas?

I have a parabola of the form $f(x)=ax^2+bx+c$ What condition must the coefficients a,b,c, meet, so that the parabola intersects the x-axis at two points? a) in the negative part of x b) on the ...
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0 votes
0 answers
21 views

Harmonic and Geometric means with Jensen inequality

Let be $a_1,...,a_n$ positive numbers, we need to proof with Jensen inequality that: $$\frac{n}{\sum _{i=1}^{n} \frac{1}{a_i} } \leq \left( \prod_{i=1}^{n} a_i \right)^{1/n} $$ Let be $x_i= \frac{1}{...
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0 answers
28 views

Find the value of 'K' .

Let $\alpha , \beta \in R$ such that $2(\alpha-1)^3+3(\alpha-1)^2+6\alpha=0$ and $2(1-\beta)^3+3(1-\beta)^2-6\beta+12=0 $ and $\alpha \neq \beta$ , also graph of $y=f(x)$ is symmetric about the point $...
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0 answers
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Are all connected graphs with degree sequence $(2,2,4,4,6)$ Hamiltonian?

Are all connected graphs with degree sequence $(2,2,4,4,6)$ Hamiltonian? I have the following few observations: Note that there are only $5$ vertices but the highest degree is $6$. Hence the graph is ...
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1 vote
0 answers
67 views

What is nature of root of the polynomial $x^5 -10x + 20?$

Problem The polynomial $x^5 -10x + 20$ has a. both positive and negative real roots b. only positive real roots c. only negative real roots d. at least two complex roots My Approach Tried to solve ...
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0 votes
2 answers
31 views

Maximizing $f(x)=\frac{1}{1+\left|x\right|}+\frac{1}{1+\left|x-1\right|}$

Find the maximum value of the function $$f(x)=\frac{1}{1+\left|x\right|}+\frac{1}{1+\left|x-1\right|}$$ For $1<x$, when $x$ increase both fractions decrease hence $f(x)$ decrease. Similarly for $x&...
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0 votes
1 answer
37 views

How do I solve for t, where $t$ is both the input to a sine function and the result?

This problem specifically, here are the steps I've taken so far: $0 = 3^{1/2} \times (1/2) \times t^{-1/2} - 2\sin(t)$ $0 = \dfrac{3^{1/2}}{(2 \times t^{1/2})}- 2\sin(t)$ $2\sin(t) = \dfrac{3^{1/2}}{...
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3 votes
1 answer
95 views

How to measure speed of a ball coming directly at you

I am trying to measure the speed of a ball based on only radius as it travels towards an observer. Assuming we don't have any x and y data for the ball. The ball is thrown directly at you at eye level ...
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2 votes
1 answer
45 views

Determining the time given the velocity and speed

Suppose a man starts to walk at a speed of $4$ km/hour. His velocity after walking $x$ km is given by $$\frac{40}{10+x}\,\text{km/hour}.$$ I am trying to determine how long he will take to walk $10$ ...
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-3 votes
3 answers
60 views

Shouldn't it be $(a/b)/c = a\cdot c/b$ instead of $a/b\cdot c$? [closed]

Shouldn't it be $(a/b)/c = a\cdot c/b$ instead of $a/b\cdot c$?
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0 answers
41 views

What's the largest sum of money using 5- cent and 7-cent coins that is not possible to create? [duplicate]

How do I know when I have hit the largest number? Is there any strategy better than guessing and checking that will work? Won't the numbers be infinitely large as there are an infinite amount of ...
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1 vote
0 answers
26 views

i'm looking at Euclid's elements i see all his propositions on ratio and proportion but im trying to figure out how it relates to our modern algebra?

could anyone point me in the direction of a resource for this, or possibly give an explanation? I'm basically wondering how the rules of algebra could be developed into what they are now.
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1 answer
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What is the algebraic rule applied to get $5\left(3^{n-1}-2^{n}\right)-6\left(3^{n-2}-2^{n-1}\right)=(15-6) \times 3^{n-2}-(10-6) \times 2^{n-1}$? [closed]

Apperently the following equivalence holds: $$5\left(3^{n-1}-2^{n}\right)-6\left(3^{n-2}-2^{n-1}\right)=(15-6) \times 3^{n-2}-(10-6) \times 2^{n-1}$$ What algebraic rule is applied to get from the LHS ...
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0 votes
1 answer
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Proving that there are infinitely many composites in an arithmetic progression [duplicate]

The question says Consider $S$={$a, a+d, a+2d, ...$} where $a$ and $d$ are positive integers. Show that there are infinitely many composite numbers in $S$ The only argument I could think of was that ...
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1 vote
1 answer
99 views

Prove that $\frac{2022}{n} + 4n$ is a perfect square iff $\frac{2022}{n} - 8n$ is a perfect square

Prove that $\frac{2022}{n} + 4n$ is a perfect square iff $\frac{2022}{n} - 8n$ is a perfect square My solution was to substitute all the positive divisors of $2022$ into the $2$ expressions and ...
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1 vote
1 answer
28 views

To find the function $g(x)$ in terms of $f(x)$.

For each point $(a,b)$ on the graph $y = f(x)$ the point $(3a-1,\frac{b}{2})$ is plotted forming the graph of another function $y = g(x)$. The problem asks us to find the function $g(x)$ in terms of $...
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1 vote
1 answer
24 views

Using Comparison test with complex numbers

So I want to show the convergence of $\sum_{k=1}^{\infty} \frac{k^{2} i^{k}}{k^{4}+1}$ using the comparison test. So I want to bound the $\frac{k^{2} i^{k}}{k^{4}+1}$ with a known number. I tried $<...
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1 vote
1 answer
52 views

For any positive integers $m$ and $n$, there are $n$ consecutive positive integers divisible by $a^m$ for some integer $a$

The problem is Prove that for any positive integers $m$ and $n$, there exist a set of $n$ consecutive positive integers each of which is divisible by a number of the form $a^m$, where $a$ is some ...
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-5 votes
0 answers
42 views

$2 x$ circumference equals area [closed]

Why does $2xπx4=25.1327412287$ and then when you multiply by two it equals $50.2654824574$ its the same as $\pi x 4 x 4$, but when I used this on a another number like 3 this strategy did not work? I ...
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0 votes
2 answers
56 views

How much gold can I buy with $2000$ cash, if the price starts at $20$ and increments by $0.01$ with each purchase?

I have problem with arithmetic. For example, $$\begin{align} \text{My cash} &= 2000 \\ \text{Gold Price} &= 20 / \text{gold} \\ \text{Increment / buying gold} &= 0.01 \end{align}$$ I would ...
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1 vote
2 answers
30 views

Other approaches to evaluate $\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}$

$$\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}=?$$ I evaluated the limit by using the Hopital rule,$$\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}=4^x\lim_{h\to0}\frac{4^h+4^{-h}-...
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  • 5,543
0 votes
1 answer
53 views

Which integrating technique should I use?

Just some context: In the mathematical course, I have undertaken this year, I've just learnt how to integrate using partial fractions, substitution(not trig though, just a variable) and integrating ...
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1 vote
0 answers
26 views

Reducible polynomial must have a factor of degree at most $n/2$

Let $f \in F[x]$ with $F$ being a field. Prove that if $f$ is reducible then it must have a factor of degree at most $n/2$, where $n\ge 2$ is the degree of $f$. My attempt: For the integral domain in ...
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