 # Master Polynomial Concepts with Comprehensive Resources

Recent questions in Polynomial arithmetic Paloma Owens 2023-03-20

## When the speed of a moving object is doubled, itsA) acceleration is doubledB) weight is doubledC) K.E. is doubledD) K.E. becomes 4 times nestalno4szl 2023-02-20

## Find the zeroes of the quadratic polynomial $4{x}^{2}-4x-3$ and verify the relationship between the zeroes and the coefficients of the polynomial. ikalawangq00 2023-01-28

## Find the sum of the first 100 natural numbers. Deven Blanchard 2023-01-10

## The highest exponent of a variable in a polynomial is called degree of the polynomial. Is this statement true or false?True of False? Gael Woodward 2022-12-26

## $\frac{{x}^{2}{y}^{-4}{z}^{3}×{x}^{-5}{y}^{5}{z}^{-6}}{{x}^{6}{y}^{2}{z}^{6}×{x}^{-4}{y}^{17}×{x}^{13}{z}^{15}}×{x}^{2}{y}^{-3}$ blogmarxisteFAu 2022-12-04

## How to find the roots for ${x}^{3}+2x+2$ Milagros Moon 2022-11-27

## A sequence is defined by the recursive function f(n+1)=f(n)−2. If f(1)=10 , what is f(3)? aplaya4lyfeSS1 2022-11-25

## Infinite series representation for root of polynomials?Given a polynomial $p\left(x\right)={a}_{n}{x}^{n}+\cdots +{a}_{1}x+{a}_{0}$, can every root of the polynomial be represented as $\sum _{k=0}^{\mathrm{\infty }}{b}_{k}$ with the ${b}_{k}$'s being a function of ${a}_{0},\dots ,{a}_{n}$ using only elementary operations of arithmetic and taking roots? blogmarxisteFAu 2022-11-24

## Finding the integral closure of k[x] in $k\left(x\right)\left(\sqrt{f}\right)$, where $f\left(x\right)={x}^{6}+t{x}^{5}+{t}^{2}{x}^{3}+t\in k\left[x\right]$Let F be a field of characteristic 2. Let $k:=F\left(t\right)$, the field of rational functions in the variable t. Let $f\left(x\right)={x}^{6}+t{x}^{5}+{t}^{2}{x}^{3}+t\in k\left[x\right]$.We want to show that the integral closure of k[x] in $k\left(x\right)\left(\sqrt{f}\right)$ is $k\left[x\right]\left[\sqrt{f}\right]$.So let $\alpha =m+n\sqrt{f}\in k\left(x\right)\left(\sqrt{f}\right)$, where $m,n\in k\left(x\right)$. Since F(t) is a field, k[x] is a PID and therefore $\alpha$ is integral over k[x] if and only if its minimal polynomial in k(x) has coefficients in k[x]. Let's then find the minimal polynomial of $\alpha$.$\begin{array}{rl}p\left(y\right)& =\left(y-\left(m+n\sqrt{f}\right)\right)\left(y-\left(m-n\sqrt{f}\right)\right)\\ & ={y}^{2}-2my+\left({m}^{2}-{n}^{2}f\right)\end{array}$and since $m\in k\left(x\right)=F\left(t\right)k\left(x\right)$ and the characteristic of F equals 2 we get that $2my=0$. Hence$p\left(y\right)={y}^{2}+\left({m}^{2}-{n}^{2}f\right)$Thus, $\alpha =m+n\sqrt{f}$ is integral over k[x] iff ${m}^{2}-{n}^{2}f\in k\left[x\right]$. I' m pretty sure that I'm on the right course but I don' t know how to continue. merodavandOU 2022-11-24

## Give an example of a cubic polynomial Emmanuel Giles 2022-11-23

## Does two's complement arithmetic produce a field isomorphic to $GF\left({2}^{n}$?From what I understand, we have these two isomorphisms:(TC,+) is isomorphic to the cyclic group $\mathbb{Z}/{2}^{n}\mathbb{Z}$.(TC,∗) is isomorphic to the multiplicative group of polynomials.If this is correct, can we conclude that two's complement arithmetic produces a finite field isomorphic to $GF\left({2}^{n}$?If not, what algebraic structure, if any, does two's complement representation and arithmetic produce? Because there just seems to be something there. Anton Huynh 2022-11-23

## Polynomial problem involving divisibility, prime numbers, monotonyLet f be a polynomial function, with integer coefficients, strictly increasing on $\mathbb{N}$ such that $f\left(0\right)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio $r>0$ such that the value of function f in every term of the progression is a prime number. I believe that the solution includes a reductio ad absurdum, but I don't know how to solve it. Zackary Diaz 2022-11-23

## Modular operation on polynomial ringsI originally started modular arithmetic by the following:1mod21/2 is 0.50 times 2 is 0$1-0=1$1 equals 1mod2.Is it the same way to compute a quotient of a polynomial ring such as $\frac{\mathbb{C}\left[{x}_{1},\dots ,{x}_{n}\right]}{{x}^{2}+{y}^{2}-{z}^{2}}$${x}^{3}+2x{y}^{2}-2x{z}^{2}+xmod\phantom{\rule{thickmathspace}{0ex}}{x}^{2}+{y}^{2}-{z}^{2}$$\frac{{x}^{3}+2x{y}^{2}-2x{z}^{2}+x}{{x}^{2}+{y}^{2}-{z}^{2}}$$\frac{\left({x}^{3}+2x{y}^{2}-2x{z}^{2}+x\right)×\left({x}^{3}+2x{y}^{2}-2x{z}^{2}+x\right)}{\left({x}^{2}+{y}^{2}-{z}^{2}\right)}$${x}^{3}+2x{y}^{2}-2x{z}^{2}+x-\frac{\left({x}^{3}+2x{y}^{2}-2x{z}^{2}+x\right)×\left({x}^{3}+2x{y}^{2}-2x{z}^{2}+x\right)}{{x}^{2}+{y}^{2}-{z}^{2}}$ Celeste Barajas 2022-11-23

## Converting a polynomial ring to a numerical ring (transport of structure)One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in parallel to ring of integers, I think converting a polynomial ring to a numerical (i.e. integer) ring is useful.What I mean by converting to a numerical ring is: in the standard ring of integers, + and ⋅ are defined as in usual arithmetic. But is there universal way of converting any polynomial/monomial rings such that each object in the ring gets converted to an integer, and + and ⋅ can be defined differently from standard integer + and ⋅? This definition would be based on integer arithmetic, though. Howard Nelson 2022-11-22

## Affine curve of an absolutely irreducible polynomialLet $f\in {\mathbb{F}}_{q}\left[X,Y\right]$ be an absolutely irreducible polynomial of degree n. Denote the affine curve defined by the equation $f\left(X,Y\right)=0$ over ${\mathbb{F}}_{q}$ by $\mathrm{\Gamma }\left({\mathbb{F}}_{q}\right)$, and let $d=\mathrm{deg}\mathrm{\Gamma }$. The for each m there exists ${q}_{0}$ such that $|\mathrm{\Gamma }\left({\mathbb{F}}_{q}\right)|\ge m$ for all $q\ge {q}_{0}$. Arendrogfkl 2022-11-21

## Showing property for the derivative ${\mathrm{\partial }}_{x}T$ of a trigonometric polynomialLet be$T=\sum _{n=0}^{N}\stackrel{^}{T}\left(n\right){e}^{inx}$a trigonometric polynomial of grade N without negative frequencies.I wanna show that${\mathrm{\partial }}_{x}T=-iN\left({F}_{N}\ast T-T\right)$Where ${F}_{N}\ast T$ meas the convolution of the Fejer Kernel and T.might be easy..but I just can't work out the right conversion for this property..SO${\mathrm{\partial }}_{x}T=\sum _{n=0}^{N}\stackrel{^}{T}\left(n\right)in{e}^{inx}$$=\sum _{n=0}^{N}\left(\frac{1}{2\pi }{\int }_{-\pi }^{\pi }T\left(y\right){e}^{-iny}dy\right){e}^{inx}in$$=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }T\left(y\right)\left(\sum _{n=0}^{N}{e}^{in\left(x-y\right)}\right)in$from there on I get carried away in the wrong direction. is the derivative right?Also..the Fejer Kernel can be expressed as the mean arithmetic value of the dirichtlet kernel so:${F}_{N}=\frac{1}{n+1}\sum _{k=0}^{n}{D}_{k}\left(x\right)$Where ${D}_{k}=\sum _{n=-k}^{k}{e}^{inx}$ is the Dirichtlet Kernel Jefferson Booth 2022-11-21

## Let $f\left(x\right)=a{x}^{3}+b{x}^{2}+cx+d$, be a polynomial function, find relation between a,b,c,d such that it's roots are in an arithmetic/geometric progression. (separate relations)So for the arithmetic progression I took let $\alpha ={x}_{2}$ and r be the ratio of the arithmetic progression.We have:${x}_{1}=\alpha -2r,\phantom{\rule{1em}{0ex}}{x}_{2}=\alpha ,\phantom{\rule{1em}{0ex}}{x}_{3}=\alpha +2r$Therefore:${x}_{1}+{x}_{2}+{x}_{3}=-\frac{b}{a}=3\alpha$${x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}=9{\alpha }^{2}-2\frac{c}{a}\to 4{r}^{2}=\frac{{b}^{2}-3ac}{3{a}^{2}}$${x}_{1}{x}_{2}{x}_{3}=\alpha \left({\alpha }^{2}-4{r}^{2}\right)=-\frac{d}{a}$and we get the final result $2{b}^{3}+27{a}^{2}d-9abc=0$.How should I take the ratio at the geometric progression for roots?I tried something like${x}_{1}=\frac{\alpha }{q},\phantom{\rule{1em}{0ex}}{x}_{2}=\alpha ,\phantom{\rule{1em}{0ex}}{x}_{3}=\alpha q$To get ${x}_{1}{x}_{2}{x}_{3}={\alpha }^{3}$ but it doesn't really work out..Note:I have to choose from this set of answers: Jefferson Booth 2022-11-20

## Integral roots for $f\left(x\right)=41$ if $f\left(x\right)=37$ has 5 distinct integral roots.Given a polynomial f(x) with integral coefficients and $f\left(x\right)=37$ has 5 distinct integral roots, find the number of integral roots of $f\left(x\right)=41$?My Approach: Say $f\left(x\right)=\left(x-{r}_{1}\right)\left(x-{r}_{2}\right)\left(x-{r}_{3}\right)\left(x-{r}_{4}\right)\left(x-{r}_{5}\right)g\left(x\right)+37$, where ri are the distinct integers.Now for $f\left(x\right)=41$ we have $\left(x-{r}_{1}\right)\left(x-{r}_{2}\right)\left(x-{r}_{3}\right)\left(x-{r}_{4}\right)\left(x-{r}_{5}\right)g\left(x\right)=4$, so the factors can be $±1,±2$ or $±4$. Given ${r}_{i}$ are distinct at most two of them will give $±1$, then there can be both of $±2$ or one of $±4$. This is where I get lost, since even if I use all of $±1,±2$, I will be still be left with one $x-{r}_{i}$ factor. What about that? Does it matter that 37 and 41 are primes, or is it just a coincidence? apopiw83 2022-11-20
## Convolution product of arithmetic functionsAn arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions f and g to be the arithmetic function $f\star g$, whereWe say that two arithmetic functions f and g are dependent if there exists a nontrivial polynomial of two variables $P\left(x,y\right)=\sum _{i,j}{a}_{ij}{x}^{i}{y}^{j}$ with real coefficients such that$P\left(f,g\right)=\sum _{i,j}{a}_{ij}{f}^{\star i}\star {g}^{\star j}=0$and say that they are independent if they are not dependent. Let p and q be two distinct primes and setProve that ${f}_{1}$ and ${f}_{2}$ are independent.My book says that${f}_{1}^{\star i}\star {f}_{2}^{\star j}\left(n\right)=\left\{\begin{array}{l}1\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n={p}^{i}{q}^{j}\\ 0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{otherwise}\end{array}$and the result follows from that, but how do they get that and how does the result follow from that? django0a6 2022-11-20