Tagged Questions
Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.
0
votes
3answers
34 views
Help solve rational expression
I need help solving this rational expression.
Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$
How do you solve this problem? Where do I start?
6
votes
1answer
37 views
Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $
If $x\in\mathbb{R}$ find the maximum value of
$$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$
I tried this:
Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$
For maxima ...
0
votes
3answers
8 views
Setting up word problem for finding length and width
Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width.
I have not idea where to start. ...
0
votes
1answer
39 views
Solving the polynominal: $s(t) = -16t^2 + 48t + 160$
The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function:
$s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
1
vote
1answer
35 views
How to find the solution of a quadratic equation with complex coefficients?
I know how to find the solution for a quadratic equation with real coefficients. But if the coefficient changes to complex numbers then what is the change in the solution? Want an example of such ...
1
vote
0answers
29 views
Kantorovich Theorem example
I need to write in C a program that finds roots of a 6th order polynomial.
I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
1
vote
1answer
52 views
I need help proving a statement about rational roots
I have no idea where to start...this is the statement:
If a polynomial of degree not greater than 5 with rational coefficients has multiple roots, it has also a rational root, except in the case ...
1
vote
0answers
27 views
Decomposition of polynomials and inequality
This was asked in comment here by @23rd :
If $f$ is a polynomial with $\deg f=n\ge2$, then there exist polynomials $g$ and $h$, such that $$f(x)=2xg(x)−h(x)$$ $$\deg g\le n−1, \quad \deg h \le ...
0
votes
1answer
44 views
Polynomial long division to determine the quotient [on hold]
Use polynomial long division to determine the quotient when 3x^3 - 5x^2 + 10x +4 is divided by 3x + 1?
0
votes
0answers
36 views
Construct a polynomial with a certain root.
Suppose we have a polynomial $g(x)=ax^3-bx^2+cx-d\in\mathbb{Z}[x]$ whose zero is $\rho$.
How do we construct a polynomial which involves $g(x)$ and with a root $\frac{d}{\rho}$?
I was trying to do ...
3
votes
4answers
389 views
Polynomials Shouldn't Have factors using Rational Root Theorem but it does!
I came across this polynomial
$X^4 + X^3 + 2X^2 + X + 1$
I tried to factor it using Rational root theorem, but it seems there are no roots possible. 1 or -1 don't work.
But I know for a fact ...
0
votes
6answers
62 views
0
votes
0answers
17 views
Sufficient condition for a indefinite integral to be an elementary function
I would like to find a sufficient condition on two polynomials $P(s)$ and $Q(s)$, such that the function $s \mapsto Q(s)e^{P(s)} $ has a primitive integral of the form $s \mapsto R(s)e^{P(s)} $ (with ...
0
votes
1answer
24 views
remainder is not zero using long division method
Find all zeros of $f(x)=128x^3-48x^2+1$ given that one linear factor
occurs twice.
let $f(x) $ be equaal to 0
$128x^3-48x^2+1=0,$
$16x^2(8x-3)+1=0,$
trying $x=1/4$
$16/16(2-3)+1=0,$
...
1
vote
1answer
59 views
Most Efficient Method to Find Roots of Polynomial
I am designing a software that has to find the roots of polynomials. I have to write this software from scratch as opposed to using an already existing library due to company instructions. I currently ...
2
votes
1answer
36 views
Determine the coefficients of a polynomial knowing its roots
My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice.
Given the following polynomial:
...
4
votes
1answer
28 views
Polynomials vanishing on subsets of $\mathbb{R}^2$
Let $\mathcal{S}\subset\mathbb{R}^2$ such that every point in the real plane is at most at distance $1$ from a point in $\mathcal{S}$. Is it true that if $P\in\mathbb{R}[X,Y]$ is a polynomial that ...
4
votes
0answers
29 views
Set of Metapolynomials is closed under multiplication
We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a
metapolynomial if, for some positive integer $m$ and $n$, it can be
represented in the form $$f(x_1,\cdots , x_k ...
6
votes
4answers
94 views
Finding double root of $x^5-x+\alpha$
Given the polynomial
$$x^5-x+\alpha$$
Find a value of $\alpha>0$ for which the above polynomial has a double root.
Here's an animated plot of the roots as you change $\alpha$ from $0$ to $1$ I'm ...
0
votes
0answers
37 views
Extension of quadratic forms
A homogen multivariate polynomial with degree 2 is a quadratic form. It can be checked
if the polynomial is positive for any non-zero vector by checking if the corresponding
matrix A is positive ...
3
votes
1answer
40 views
What is the Most Efficient Way to Calculate the Internal Rate of Return?
I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to.
I currently use the ...
3
votes
0answers
72 views
An integral and $\pi(n)$
Are there polynomials $P,Q\in \mathbb{R}[x]$ satisfying :
$$\int_{0}^{\log n}\frac{P(x)}{Q(x)}\,\mathrm{d}x=\frac{n}{\pi(n)}\quad \text{ for infinitely many }n\in \mathbb{N}$$
Here $\pi(n)$ is the ...
0
votes
5answers
44 views
Fraction with negative exponent fraction.
Q:
$$\left(\frac{27 a^6 b^{-3}}{c^{-2}}\right)^{-2/3}$$
A: $$\frac{b^2}{9 a^4 c^{4/3}}$$
How in the world are they getting that?
0
votes
0answers
15 views
evaluation of polynomial regression
I have a data set $(x_i$ $y_i)$ if=1...20. I have to fit the data using polynomial feature. How can I evaluate what the complexity of model should be chosen?
There is a hint in the task using RMSE ...
2
votes
1answer
22 views
Explicit delta for polynomial limit
I'm looking for an explicit formulation for $\delta$ in the $\epsilon-\delta$ formulation of the limit for a polynomial $p(x) = \sum_{n=0}^N a_nx^n$.
For example, in the the specific linear case ...
-1
votes
1answer
57 views
Finding a value that a set of given polynomials have the largest possible common divisor
Let $f_1, ..., f_n$ be homogenous polynomials in $Z[x_1, .., x_4]$. Find the value $\alpha=(\alpha_1, ... , \alpha_4)$ such that $f_1, ... , f_n$ evaluated at $\alpha$ have the largest common ...
0
votes
3answers
83 views
for which a, the matrix A is diagonalizable?
A = $
\begin{pmatrix}
2a+3 & 0 & 0 \\
-a-3 & a & a+3 \\
a & a & a+3 \\
\end{pmatrix}
$
Characteristic polynomial:
$
...
1
vote
0answers
15 views
Derivation of composite Gaussian quadrature error formula
I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following:
Derive the error term for the composite Gaussian quadrature rule with ...
4
votes
0answers
65 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 $\mathbb{Q}[x]$.
1
vote
1answer
77 views
How to Show Polynomial Growth < Exponential Growth (Without L'Hopital!)
Can anyone offer me a way to show that exponential growth trumps polynomial growth, without using L'Hopital's Rule? When I learned function growth speeds in high school, the closest thing to a proof I ...
2
votes
0answers
39 views
How “separable” (not in that sense) is a polynomial?
Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
0
votes
3answers
42 views
In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$…Using Euclid's Method. [closed]
In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$...Using Euclid's Method.
Could someone please do the first few steps of this so I know how to solve gcd of polynomials using Euclid's ...
0
votes
2answers
43 views
Representing a higher degree polynomial as product of smaller degree polynomials?
Consider an equation
$H(Z)=1+\frac 52Z^{-1}+2Z^{-2}+2Z^{-3}$ I want to write it as a product of a first degree polynomial and another polynomial, which will be...$$H(Z)=(1+2Z^{-1})(1+\frac ...
0
votes
0answers
21 views
vector subspace of all real polynomials which are divisible with $x^2 + 1$
Show that the set of all real polynomials which are divisible with $x^2 + 1$
is a vector subspace of space of all real polynomials to 4th degree.
Also find base and dimension of this subspace.
I ...
0
votes
3answers
59 views
complex roots calulation question
How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
1
vote
2answers
84 views
Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$
Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$.
I'm having problems finding the roots...this is what I've done:
First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
3
votes
0answers
29 views
Determining if a given equation is solvable given a set of ultra-radicals
So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots)
AS WELL AS a set of inverses for some polynomials which are not solvable using ...
1
vote
1answer
44 views
About $\mathbb{F}_7[x]$
can you help me with this?
Let $a(x)=3x^6+2x^2+x+5$ and $b(x)=6x^4+x^3+2x+4$, find the g.c.d between $a(x)$ and $b(x)$ in $\mathbb{F}_7[x]$.
Thanks!
2
votes
2answers
214 views
Finding the remainder when a polynomial is divided by another polynomial. [duplicate]
Find the remainder when $x^{100}$ is divided by $x^2 - 3x + 2$.
I tried solving it by first calculating the zeroes of $x^2 - 3x + 2$, which came out to be 1 and 2.
So then, using the Remainder ...
0
votes
2answers
27 views
Finding a cubic polynomial whose zeroes are the same as collectively of two other quadratic polynomials.
The question is:
Find a cubic polynomial $p(x)$ whose zeroes are the same as those collectively of polynomials $g(x) = 2x^2 - 9x + 4$ and $f(x) = 2x^2 + 3x - 2$. Given that $p(0)$ = 8.
I tried ...
1
vote
1answer
18 views
Finding the remainder polynomial for a given polynomial.
When a polynomial $p(x)$ of degree 3 is divided by $3x^2 − 8x + 5$, quotient and remainder obtained are linear polynomials such that $p(1)$ = 19 and $p(5/3)$ = 25. So, find the remainder polynomial.
...
-4
votes
1answer
60 views
A bunch of questions involving polynomials.
Okay, so apparently, I can't write more than one post in the space of 20 minutes. So, I'm writing down all the questions I wasn't able to solve here.
It would be great if you could solve them and ...
0
votes
2answers
35 views
Real polynomials, complex zeroes and the Intermediate value theorem
I have a second grad polynomial p(x). For arguments sake lets say
$$p(x) = x^2 + 16x + 76$$
I also have an inequation
$$p(x) > 0$$
Now the inequation does not have a real solution, but only ...
0
votes
2answers
45 views
Find the values of a,b and c in a polynomial $p(x) = ax^2 + bx + c$
The question is this :
A polynomial $p(x) = ax^2 + bx + c$ where $a,b,c$ are some rational numbers, has $1 + \sqrt3$ as one of the zeroes and also $p(2) = -2$. Find the values of $a,b$ and $c$.
...
1
vote
2answers
103 views
Find the value of “k” so that the quadratic polynomial has equal zeroes.
The question is this:
Find the the value(s) of $k$ so that the quadratic polynomial $kx^2 + x + k$ has equal zeroes.
Answers along with appropriate explanations would be appreciated.
Thanks.
3
votes
3answers
512 views
What is the lowest-degree function that passes through these points?
I want to find a (preferably polynomial) function that passes through the following twelve points:
$(1, 0)$
$(2, 3)$
$(3, 3)$
$(4, 6)$
$(5, 1)$
$(6, 4)$
$(7, 6)$
$(8, 2)$
$(9, 5)$
$(10, 0)$
$(11, ...
3
votes
2answers
32 views
Build field extension and solve equation
Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field.
As I understand we need to build $\mathbb{F}_{5^{2}}$.
Field $\mathbb{F}_5$ contains ...
0
votes
2answers
60 views
Matrix with rank 3 does not exist in this $p(x)$
Given: Characteristic polynomial is $p(x) = x^7 - x^5 + x^3$ .
Prove that there isn't a matrix A that $ \rho(A) = 3 $
I tried to play with $p(x) = x^3(x^4 - x^2 +1)$ But I'm still not sure how ...
0
votes
1answer
42 views
Polynomials - getting wrong answer using Euclidean algorithm
I am finding the GCD of $a = x^3 + 11/3x^2 + 17/4x + 3/2$ and $b = 3x^2 + 22/3x + 17/4$ using the Euclidean algorithm. So I divide $a/b$ and get $q$ and $r$ such that $a = qb + r$. Then, according to ...
4
votes
2answers
61 views
Need help with a diophantine expression
I'm faced with this problem. Under what conditions is this expression a positive odd integer:
$$\frac{2^g(x^2+y^2-z^2)}{x+y-z}$$
where $g,x,y,z$ are nonnegative integers. x and z are odd, and y is ...
