Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, dividing, factoring and solving for roots.
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1answer
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Solving 3 simultaneous cubic equations
I have three equations of the form:
$$ i_1^3L_1 + i_1K +V_1 + (i_2+i_3+C)Z_n = 0 $$
$$ i_2^3L_2 + i_2K +V_2 + (i_1+i_3+C)Z_n = 0 $$
$$ i_3^3L_3 + i_3K +V_3 + (i_1+i_2+C)Z_n = 0 $$
where $ ...
0
votes
1answer
40 views
Best way to write the characteristic polynomial
In linear algebra I've seen that there are two major different ways
to write the characteristic polynomial of a mapping $f$: As
$$
...
0
votes
1answer
20 views
Minimal polynomials over the rationals and the reals
Find the minimal polynomial over $\mathbb Q$ and $\mathbb R$ for ...$\sqrt[3]{3}$, $1- i\sqrt{3}$, $2 + i$, $i\sqrt[3]{3}$
Sorry for my sqrt formulas .. I'm new here, hope to learn really fast to ...
0
votes
0answers
24 views
Question on Polynomials
I have to determine the value of a $\in \mathbb{R}$, if any such that
$$\sum_{x=1/2}^{15/2} (a + 2x)^{[8]} =0$$
Ordering = ${1 \over 2}, {3 \over 2},{5 \over 2},... ,{15 \over 2}$
Note: To define ...
4
votes
2answers
39 views
Check whether the following polynomial is irreducible over $\mathbf Q$
I was trying this problem from my Abstract Algebra book exercise that says:
Show that the polynomial $x^2+\frac 13x-\frac 25$ is irreducible in $\mathbf Q[x]$.
What I tried: $x^2+\frac ...
0
votes
1answer
23 views
Resultant of two polynomials f and g
I know how to calculate the resultant of two polynomials.. but I am little confused with an example I have come across in a past paper...
The type of example I am used to are f= $5X^3-185X-420$ and ...
1
vote
3answers
89 views
Solving the equation $\dfrac{(1+x)^{36} -1}{x} =20142.9/420$ for $x$.
How would one solve for x in the following equation:
$\dfrac{(1+x)^{36} -1}{x} =20142.9/420$
I tried factorising the top but that didnt really help much.
$((1+x)^{18} - 1)((1+x)^{18}+1)$
Any help ...
3
votes
2answers
31 views
sum of polynoms of given property
I have $P(x)$ a polynomial with degree $n$ ,$P(x) \ge 0$ for all $x \in$ real.
I have to prove that:
$f(x)=P(x)+P'(x)+P"(x)+......+P^{n}(x) \ge 0$ for all $x$.
I tried different methods to ...
1
vote
1answer
34 views
Problem to understand Hungerford's book
In the Hungerford's book and following the answers of this question:
Help to understand the ring of polynomials terminology in $n$ indeterminates I have troubles to understand the following remark in ...
1
vote
4answers
28 views
Prove that if $p(A)=0$ where A is a matrix of a linear operator ($A \in L(V)$), $p(\lambda)=0$ if $\lambda \in \sigma(A)$
I think it's all in the title. $p$ is some random polynomial.
I don't know how to approach this one. I've tried taking the roots of $p$, placing them on the diagonal of a new matrix and reasoning that ...
1
vote
1answer
33 views
Find the 2nd-degree polynomial that approximates with the method of the least squares the:$f(x)=\frac{1}{10}x^2-2x+10$
It is known that a rectangular set of polynomials $\phi_k(x), k=0,1,\cdots,n$ for each $x\in[a,b]$ as to a weight function $w(x)$ can be constructed with the use of the following recursive type ...
0
votes
0answers
24 views
conditions on coefficients of univariate polynomial so that it has only real roots
Consider a univariate polynomial of degree $n$ with real coefficients. Are there general equalities/inequalities on its coefficients, so that it has precisely $n$ real roots?
For example for the case ...
0
votes
0answers
42 views
Proved ideal generated by… Is a monomial ideal
Similar questions have come up on the last few past exam papers and I don't know how to solve it.. Any help would be greatly appreciated..
Prove that the ideal generated by $\mathbb{Q}[X,Y]$ ...
3
votes
3answers
51 views
Find $p$ if $(x + 3)$ is a factor of $x^3 - x^2 + px + 15$.
I'm just making sure I answered this correctly.
If $(x+3)$ is a factor, then $P(-3)$ would equal $0$, correct?
0
votes
3answers
39 views
How to graph the equation: y=(x-2)/(x+1)?
the title says it all.
I'm pretty sure this is a hyperbola, but is there an alternative way of doing this besides a table of values?
"Graph the equation $y=(x-2)/(x+1)$"
I know that $x$ cannot equal ...
1
vote
2answers
28 views
Help to understand the ring of polynomials terminology in $n$ indeterminates
In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner:
After the author defines the operations in this ring with a theorem:
...
2
votes
1answer
25 views
For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
4
votes
2answers
74 views
What am I doing wrong in these quartic formula calculations?
I was a bit surprised that there is a general formula for the roots of a quartic equation, so I decided to test Wikipedia's version of it myself. To my surprise, I have arrived at a correct answer ...
1
vote
1answer
21 views
Finding the value of a coefficient given the roots are equal?
I've had some trouble with this question:
"$P(x)$ denotes the quadratic polynomial $kx^2+(k-1)x-(2k-1)$, where $k$ is a rational, real number. Find the value of $k$ for which the roots of $P(x)=0$ ...
1
vote
3answers
48 views
find out the value of $\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$
If $(x-3)^2+(y-5)^2+(z-4)^2=0$,then find out the value of $$\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$$
just give hint to start solution.
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votes
2answers
14 views
Deducing a coefficient from a cubic polynomial given a divisor and remainder?
I got this question which I don't understand:
"Suppose $x^3 - 2x^2 + a = (x + 2) Q(x) + 3$ where $Q(x)$ is a polynomial. Find the value of a."
I know the identity: $P(x)=A(x)Q(x)+R(x)$, but I'm not ...
1
vote
0answers
43 views
A question about cubic equation.
I'd like to share my doubt on cubic equation.
Step 1: $ax^3+bx^2+cx+d=0$,
Step 2: We can substitute $x=y-\frac b {3a}$ to get $y^3+py+q=0$ where $p,q$ are something.
Step 3: By Vieta's ...
4
votes
2answers
59 views
Simplifying this expression $(e^u-1)(e^u-e^l)$
Is it possible to write the following
$$(e^u-1)(e^u-e^l)$$
as
$$e^{f(u,l)}-1?$$
4
votes
3answers
47 views
Factorize in R[x]
I have the polynomial $x^8+1$, I know that there's no root for solve this in $\Bbb R[x]$ but i want to factorize this to the minimal expression. This is possible or this is irreducible?
3
votes
5answers
101 views
Polynomials - The sum of two roots
If the sum of two roots of
$$x^4 + 2x^3 - 8x^2 - 18x - 9 = 0$$ is
$0$, find
the roots of the equation
4
votes
4answers
64 views
Irreducible Polynomial in $\mathbb F_{256}$.
Let $\mathbb F_{256}$ be the finite field with $2^8 = 256$ elements. Consider the polynomial over this field
$$
x^2 + x + 1.
$$
I wanted to know if it is irreducible, so I calculated it for all ...
0
votes
0answers
15 views
Information about a particular polynomial
This question is related to this post Consider the following polynomial
$$
\alpha^3 xyzw + \alpha^2(1-\alpha) yzw + \alpha(1-\alpha) zw + (1-\alpha)w.
$$
One can of course generalize this to $n$ ...
0
votes
1answer
21 views
(Follow Up) Checking the solutions of a quadratic polynomial
I'm following up from this question:
Solve a polynomial involving geometric progression?
I have had trouble with this question:
"Solve the equation $8x^3−38x^2+57x−27=0$" if the roots are in ...
0
votes
2answers
36 views
Solve a polynomial involving geometric progression?
I have had trouble with this question:
"Solve the equation $8x^3 - 38x^2 + 57x -27 = 0$" if the roots are in geometric progression.
Any help would be appreciated.
1
vote
2answers
49 views
How to prove two polynomials have no zeroes in common?
The question asked:
Divide the polynomial $P(x) = x^3 + 5x^2 - 22x - 6$ by $G(x) = x^2 - 3x + 2$. I did, and got the answer: $(x+8)(x^2-3x+2)-22$.
However, it now asks to: "Show that $P(x)$ and ...
2
votes
3answers
24 views
How to form a cubic equation with the substitution method?
I had this question:
"Find the cubic equation whose roots are twice the roots of the equation $3x^3 - 2x^2 + 1 = 0$"
In my first attempt, I solved it through the use of simultaneous equations, where ...
1
vote
1answer
19 views
solving for one variable in terms of others
A question from Steward's Precalculus textbook 5th, Pg 55,
the original formula is $$h=\frac{1}{2}gt^2+V_0t$$
the question asks to write the formula in terms of $t$, the answer is ...
2
votes
0answers
31 views
Find the factorization of the polynomial as a product of irreducible [duplicate]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$
Testing with the simplest possible root in this case, $P(1)=0$
Applying the ...
1
vote
0answers
40 views
How to use two number to form a Jones polynomial
According to the Wikipedia article on Knots,
The number of crossing (rule $1$) and a line crossing the triangle (rule $2$) form a number such as $3,1$. With these two numbers, how do you form a ...
5
votes
6answers
221 views
Cubing a simple thing
I am trying to expand $\quad (x + 2)^3 $
I am actually not to sure what to do from here, the rules are confusing. To square something is simple, you just foil it. It is easy to memorize and execute. ...
1
vote
1answer
36 views
Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$
Testing with the simplest possible root in this case, $P(1) = 0$
...
2
votes
2answers
33 views
Deducing a coefficient from a cubic polynomial?
I fully answered the question, and got that $k=-3$, but the answer says it's positive. Can anyone show me my mistake?
"Given that $x-2$ is a factor of the polynomial $x^3 - kx^2 - 24x + 28$, find $k$ ...
6
votes
1answer
51 views
Calculating in quotient ring of $\mathbb{R}[X]$
Part of an old Oxford exam (1992 A1)
We want to find which elements of the quotient ring $\mathbb{R}[X]/(x^3-x^2+x-1)$ are equal to their own square.
Now, we note first that ...
3
votes
2answers
55 views
How to show that there does not exist any integer $b$ with $f(b)=14.$
Let $f(x)$ be a polynomial with integer coefficients. Suppose that there exist distinct integers $a_1,a_2,a_3,a_4,$ such that $f(a_1)=f(a_2)=f(a_3)=f(a_4)=3.$ Then show that there does not exist ...
2
votes
1answer
35 views
Minimal Polynomials Annihilating an Abelian Torsion-Free Group
Let $A$ be an abelian torsion-free group. Let $\theta \in\operatorname{Aut}A$. Assume that $\theta$ has a finite period in $\operatorname{Aut} A$, say $n$. Obviously $\theta^n-1$ annihilates $A$ (i.e. ...
0
votes
1answer
11 views
Polynomials -finding sum of symmetric function of cubic polynomial
Problem :
If $\alpha , \beta,\gamma$ are the roots of $x^3+bx+c=0$ then $\alpha^2\beta +\alpha \beta^2+\beta^2\gamma +\beta \gamma^2+\gamma^2 \alpha+\gamma \alpha^2$ is equal to
Options are :
a) ...
0
votes
1answer
30 views
Coefficients of Newton interpolation polynomial
Given distinct $y_0,...,y_m$ in $\mathbb R$, let $N_m(x)$ be the Newton interpolation polynomial of degree $m$. That is, $N_m(x) = \sum_{n=0}^{m}a_nw_n(x)$ where $w_0 = 1$, $w_n(x) = ...
4
votes
2answers
95 views
Irreducibility of $x^n-x-1$ over $\mathbb Q$
I want to prove that
$p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible.
My attempt.
GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
0
votes
0answers
21 views
Number of roots to system of Polynomials
If you have a system of k polynomials of dimension k and degree r is the number of solutions equal to:
k^s?
This appears to be the pattern and intuitively one could argue that each of the s systems ...
1
vote
0answers
20 views
Can you help me reverse the Minimum Curvature Method?
The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
6
votes
1answer
71 views
Bounding the roots of the sum of two monic polynomials with real coefficients.
Let $P_1(z)$ and $P_2(z)$ be monic polynomials with real coefficients and roots $\{z_1^{(1)},z_1^{(2)},...\}$ and $\{z_2^{(1)},z_2^{(2)},...\}$, respectively. Are there any results relating the ...
6
votes
2answers
74 views
Zeros of a cubic polynomial with rational coefficients
While discussing a related problem, one of my friends came out with a question as follows:
Is it possible that a cubic polynomial $p(x) \in \Bbb{Q}[x]$ has all of its zeros to be both real and ...
1
vote
2answers
36 views
What is the general equation of a cubic polynomial?
I had this question:
"Find the cubic equation whose roots are the the squares of that of $x^3 + 2x + 1 = 0$"
and I kind of solved it. In that my answer was $x^3 - 4x^2 + 4x + 1$, but it was actually ...
2
votes
1answer
51 views
Is there a geometric relationship between plane geometry and polynomials?
It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a ...
1
vote
2answers
66 views
A simple proof about $e^x$?
Do you guys think this is correct? I am trying to prove that there is no single-term polynomial function (oxymoron, I know) $f(x)$ which is always (or at least as x approaches infinity) greater than ...
