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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2
votes
2answers
23 views

Problem related to remainder

A polynomial in $x$ leaves a remainder $2$ and $3$ when divided by $x-1$ and $x+1$. What is the remainder, when divided by $x^2-1$ ?
0
votes
0answers
8 views

consider if the number is negative or positive

We have number $\displaystyle a= - \sum_{1\le}\sum_{i<j}\sum_{<k\le n} a_ia_ja_k + \sum_{1\le}\sum_{i<j}\sum_{<k < p}\sum_{\le n} a_ia_ja_ka_p - ... +/- a_ia_ja_ka_p...a_n $ When $n$ ...
3
votes
0answers
30 views

Easy Derivates / Rate Of Change Help

It's been many years since I've done any kind of math, so I spent a while studying it again. I have this simple question but the numbers seem really weird so I'm not sure if I have correctly solved ...
8
votes
4answers
266 views

Is there any good reason not to define $0^0=1$ , such as contradictions in algebra or arithmetic?

Math people: The title is the question: Is there any good reason not to define $0^0=1$ , such as contradictions in algebra or arithmetic? I searched for similar questions before I posted this ...
0
votes
2answers
9 views

Variable with an exponent variable

I'm actually dealing with an economics problem, but it seems like the math is always what messes me up. Ignoring what the variables mean, I'm trying to understand how to get from step 1 to step 2. $$ ...
1
vote
5answers
146 views

Algebra problem stumping me

I have recently run into an algebra problem that goes as follows. Using the digits $1$ to $9$, $$ \left\{ \begin{align} A + B + C + D &= EF \\ E + F + G + H &= CJ \\ B + G + J ...
0
votes
1answer
24 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
3
votes
3answers
57 views

Nice parameterization of $x^2 + y^2 - kx^2y^2 =1$

Can anyone find a nice simple parameterization of this curve. Just the quarter where $x \ge0$ and $y \ge0$ would be fine. The parameterization should be "nice" in the sense that the first derivative ...
1
vote
2answers
142 views

Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$

Note: this is a programming challenge at this site For this equation $$\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}\quad ( N \text{ factorial} ),$$ find the number of positive integral solutions for ...
0
votes
1answer
20 views

Finding two unknowns from equation [on hold]

I have $$\frac{P(\frac{L}{2})^3}{6}+C_1(\frac{L}{2})=\frac{PL(\frac{L}{2})^2}{4}+C_4$$ And in my textbook the next step states that $$C_1=-\frac{3}{8}PL^2$$ $$C_4=-\frac{11}{48}PL^3$$ But they do ...
3
votes
4answers
3k views

Factoring 4 term polynomial

Trying to figure this one out but I see no logical approach to this at all. $x^3-3x^2-4x+12$ I know that it will be 3 parts most likely and that each will start with x but beyond that I will just ...
0
votes
2answers
18 views

Simplify ((L/2)^3)/6

I have the expression $\frac{({\frac{L}{2}})^3}{6}$ but do not know the steps to simplify. Can someone please explain the steps for simplifying this. Thanks
3
votes
4answers
14k views

Finding the equation of the normal line

I have a question to find the equations of the tangent line and the normal line to the curve at the given point. I can find the equation for the tangent line easily but I am not sure what a normal ...
0
votes
2answers
16 views

simplifying 3 different equations with 3 different variables

I am stuck trying to get the values for x, y, and z. I keep moving variables around but I end up getting answers like x = x or z = z and I do not think that is what I want. It's really just algebra ...
2
votes
4answers
80 views

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$ I'm not sure how to properly deal with this function and solve for $f(x-2)$.
1
vote
2answers
38 views

Minimum value of a function

For $x \in [0, 5]$, let $$f(x) = \sum_{i = 1}^{5}\frac{1}{|x - i|}.$$ Why is $$f(x) \geq 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = f(0)?$$ This of course is true if one simply plots ...
11
votes
6answers
367 views

Elementary proof that $\pi < \sqrt{5} + 1$

I wanted to show that $$ \frac{\pi}{4\phi} < \frac{1}{2} $$ Where $\phi$ is the golden ratio. I have confirmed the results numerically, and by simple algebra the inequality simplifies down to $$ ...
0
votes
0answers
14 views

Descartes rule of signs

I'm trying to write an algorithm that gets a polynomial and gives how many roots does it have in the interval [0 $x_0$]. I'm supposed to do it by Descartes law, I know that by Descartes law you Know ...
3
votes
3answers
100 views

which of $\sqrt{5}+\sqrt{13}$ or $ \sqrt{34}$ is larger?

which of $\sqrt{5}+\sqrt{13}$ or $ \sqrt{34}$ is larger? I tried to define $f= \sqrt{x}+\sqrt{x+8}+\sqrt{x+29}$ and use cslculus but have failed. please helps. Is there any non calculus solution? ...
0
votes
1answer
18 views

Word problem — water pipes and pool

Three pipes 1,2 and 3 fill together a pool in 6 minutes. Pipe 2 alone fills the pool in 75% of the time that pipe 1 alone fills a pool. Pipe 3 alone fills a pool in 10 minutes longer than pipe 2 ...
0
votes
2answers
39 views

$-5| 2+4x | = -32(x+3/4)- | x | + 1$

This was my attempt: $$-5| 2+4x | = -32\left(x+\frac34\right)- | x | + 1\\ \implies|2+4x|=\frac{-32x-24- | x | + 1}{-5}\\ \implies2+4x=\pm \frac{-32x+-24- x + 1}{-5}\\ \implies4x=\pm \frac{-33x+-23 ...
0
votes
6answers
361 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,
0
votes
1answer
16 views

Converting shares from the chemical disassociation equation into fractions

Some basic math is eluding me when trying to derive a simple disassociation constant formula. Given that $K_d=\frac{[A][B]}{[AB]}$, $[A]+[AB]=[A_0]$, $[B]+[AB]=[B_0]$, and $[B_0] \gg [A_0]$ I'm ...
-1
votes
4answers
167 views

Adding fractions is not at all obvious

Why does $\frac{5}{4} + \frac{2}{3}$ need to be rewritten as $\frac{15}{12} + \frac{8}{12}$ to be added? It's not obvious. I'm looking towards the fact that any integer can be rewritten as $x=qy$ ...
39
votes
15answers
33k views

What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We are factoring polynomials and we both had the same thought at the same time: when are we going to use this? I feel ...
16
votes
8answers
3k views

Kid's homework: 4 equations 5 unknowns? Going crazy!

I'm new here, and I'm hoping someone can help out. My 10 year old son has been set a maths problem, which I can't solve. I've got a PhD in neuroscience and do a fair amount of matlab stuff (data ...
1
vote
0answers
25 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
0
votes
1answer
21 views

Polar graph question

Can you only graph periodic functions using polar graphing? I'm not really understanding this I guess. It you are to get all of the x and y values on a finite graph, then the original must be ...
0
votes
1answer
10 views

Linear programming problem answer

Big Seas makes regular ice cream and non fat ice cream. The ice cream mixer can make at most 300 gallons. Each regular ice cream requires 5 ounces of milk, and each non fat ice cream requires 2 ounces ...
0
votes
1answer
15 views

What expression represents the total cost?

A customer calculated the cost of a new jacket , c, including a 7% sales tax, by multiplying 0.07 times the cost of the jacket and adding the product to the cost of the jacket. What is another way to ...
0
votes
0answers
23 views

Distance between point A and and point B.

A surveyor on one side of a river wishes to find the distance between points A and B on the opposite side of the river. On her side, she chooses points C and D, which are CD = 20 m apart, and ...
1
vote
1answer
26 views

Would every half angle of an angle in each quadrant be in the previous quadrant?

For example, take (5pi)/4 which is in Q3, it's half angle is (5pi)/8 which is in Q2. Is this true for every angle?
0
votes
2answers
16 views

How many ounces of water is necessary to dillute an active ingredient in a solution?

Solution A has 10% active ingredient and 90% inactive ingredient. You have exactly 0.25 ounces of solution A. How many ounces of water must be added to solution A to dilute the active ingredient to ...
5
votes
2answers
85 views

A different type binomial expansion problem

Suppose we have $$(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}.$$ What will be the value of $a_0^2 - a_1^2 + a_2^2 - \cdots + a_{2n}^2$? The answer is $a_n$, but I can't solve it. ...
2
votes
1answer
56 views

$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $?

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $ where $m,k \in \mathbb N$ ? I tried it with a few numbers and it seems to work every time.
5
votes
2answers
88 views

Solving complex trig functions: $\sin2x + \sin3x = \frac{\sqrt{3}}2$

How to solve: $$\sin(2x) + \sin(3x) = \frac{\sqrt{3}}{2}$$ where $x$ is in $[-\pi,\pi]$? I have no idea what to do with the $\sin(2x) + \sin(3x)$. Am I supposed to factorise, differentiate, is ...
0
votes
2answers
67 views

How to solve this inequality?

I have the following problem in an assignment and have been struggling to do it. $2 + 2x - x^2 \geq 2 \sqrt{1+2x}$ I have tried solving for $x$ but have not been able to do so. Any hints to solve ...
2
votes
6answers
49 views

General solution for squared trigonometry questions: $\cos^2 x = 1$

$\cos^2 x = 1$ How do you solve trig equations with a power? Unsure what to do with the square? I get this $\frac{1+\cos2x}2 =1$ $\cos2x =1$ $2x=2n\pi\pm0$ $x=n\pi$ but the answer says $\pm ...
0
votes
2answers
69 views

How many days are there in 70 years?

How to calculate the total no. of days in 70 years (or any other no. of years) considering that this period also includes leap years?
2
votes
2answers
84 views

If $x_1^3+x_2^3+\ldots+x_t^3=2002^{2002}$, find minimum value of $t$ so the condition can be satisfied by some natural numbers $x_i$

If $x_1^3+x_2^3+\ldots+x_t^3=2002^{2002}$, find the minimum value of $t$ so the condition can be satisfied by some natural numbers $x_i$. My attempt: I took modulo $9$ on both sides and found the ...
0
votes
1answer
102 views

Proof for A majorizes B

$\alpha = [\alpha_i] \in\mathbb R^n$ and $\beta = [\beta_i]$ where $\beta_1 = \beta_2 = ......=\beta_n = \frac{1}{n}\sum\alpha_i$ How can i show that $\alpha$ majorizes $\beta$ I tried to get a ...
3
votes
3answers
90 views

Absolute Value inequality help: $|x+1| \geq 3$

Find the solutions to the inequality: $$|x+1| \geq 3$$ I translate this as: which numbers are at least $3$ units from $1$? So, picturing a number line, I would place a filled in circle at the ...
0
votes
2answers
81 views

Prove $|x+1|\leq 4$ implies that $-4\leq x\leq 2$.

How do I prove that if $x$ is a real number, then $\lvert x+1 \rvert\leq 3$ implies that $-4\leq x\leq 2$. EDIT: $\lvert x+1 \rvert\leq 4$ should be $\lvert x+1 \rvert\leq 3$
16
votes
16answers
14k views

Does .99999… = 1?

I'm told by smart people that 0.999... = 1 and I believe them but is there a proof that explains why?
0
votes
1answer
721 views

Find the common ratio of geometric progression

If $p,q$, and $r$ are terms of an arithmetic progression are also in a geometric progression, then find the common ratio of the geometric progression in terms of $p,q$, and $r$.
0
votes
1answer
34 views

How does a sequence's convergency change finite sums?

What has been troubling me lately is that I cannot grasp how a finite series could ever diverge if a finite sequence that is divergent can only imply to a finite sum every time. Perhaps my main ...
1
vote
0answers
17 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
0
votes
3answers
93 views

Prove that $ x^n - y^n = (x-y) (x^{n-1}+x^{n-2}y\,+ \,\,…\,\,+ y^{n-1})$ [closed]

Prove that $ x^n - y^n = (x-y). (x^{n-1}+x^{n-2}y\,+ \,\,...\,\,+ y^{n-1}) $; $\,\,\,\,\,$$x,y \in \mathbb{R}$
5
votes
0answers
42 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
3
votes
5answers
107 views

Domain of $\sqrt{1+\frac1x}$

I solved this as follows: $1+\frac1x \ge 0$ $\frac1x \ge -1$ (subtract $1$ from both sides) $1 \ge -x$ (multiply $x$ to both sides, cancel out $x$ from bottom of left side) $-1 \le x$ (multiply ...