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.

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3
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3answers
45 views

Conditions for distinct real roots of cubic polynomials.

Given a cubic polynomial with real coefficients of the form $f(x) = Ax^3 + Bx^2 + Cx + D$ $(A \neq 0)$ I am trying to determine what the necessary conditions of the coefficients are so that $f(x)$ has ...
-2
votes
2answers
98 views

Grade 7: When is an Algebraic Expression NOT a Polynomial? [duplicate]

Ok so this appeared in our school, and we all didnt know the answer. We arent sure, but we think the answer is, "when there are Signs like the Square Root signs, Negative Integers, etc." Or "When a ...
-1
votes
1answer
25 views

Quadratic problems [closed]

For the equation $f(x)=x^5+ax^4+bx^3+cx^2+dx-420$, why do we use $f(x)=(x-\alpha)(x-\beta)(x-\gamma)(x-\delta)(x-\mu)$ with multiplication of the roots giving$\alpha\beta\gamma\delta\mu=420= 2\times ...
-4
votes
2answers
42 views

Making a Perfect square [closed]

How to get max value of $-2x^2+3x+5$ by making perfect square. I got wrong ans. Correct ans. Must be $49/8$.
0
votes
0answers
19 views

Plot zeroes of polyomial in $\mathbb{R}[x,y,z]$ in Linux

I have a polynomial in $\mathbb{R}[x,y,z]$, say $$p_c(x,y,z)=x^2+y^2+z^2-xyz-c.$$ Do you know any simple piece of software to plot the zeroes of this polynomial using Linux? In the past I plotted ...
1
vote
1answer
30 views

Derivative of polynomial in GF(16)

how can I find the derivative of the following polynomial in $GF(2^4)$: $\alpha x^4+x^3+\alpha x^2+\alpha^2 x+1$ ?
-1
votes
2answers
87 views

Need help with an Elementary Math question [closed]

If $a+b+c=1$ and $ax^2 + bx + c = 0$ has a unique solution. Find $a,b$, and $c$.
8
votes
0answers
210 views

How to simplify this combinatorial expression?

Find \begin{eqnarray} \sum_{j\in\mathbb{N}}(n-2j)^k\binom{n}{2j-m} \end{eqnarray} Note that this question is a generalization of this one. I tried to imitate the steps in the answer given in that ...
0
votes
1answer
46 views

Finding real roots of a Polynomial Equation without graphs.

I am interested in finding the number of real roots of this polynomial : $x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} = 0$ Okay, I know that graphing it would tell me how many real ...
1
vote
1answer
29 views

Find all polynomials $P \in \mathbb{R}[x]$

Find all polynomials $P \in \mathbb{R}[x]$ such that $$P(x^2-2x)=P(x-2)^2.$$ I think we should replace $x$ with $x+1$, I really don't know, any help?
0
votes
0answers
30 views

Integral roots of a polynomial

I have one doubt. Suppose, $f_{n}(x)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+,...,+a_{n-1}x+a_n=0$ be a polynomial with an integral coefficients. If for some $n$ ( say $n=2 \ or \ 3$) , $f_{n}(t)=0,$ where, $t ...
3
votes
1answer
59 views

Non-linear equivariant maps between group representations

Given two representations $\pi_1$ and $\pi_2$ of a group $G$ (let's say it's a compact Lie group), a natural thing to study are linear equivariant maps A between them: $$ A \pi_1 = \pi_2 A $$ I'm ...
1
vote
1answer
36 views

semi definite representation of a polynomial

Let $P\left( x \right) = {a_0} + {a_1}x + ..... + {a_n}{x^n}$ be a polynomial with degree $n$. How can I write the vector ${\bf{a}} = [{a_0},....{a_n}]$ in a semidefinite matrix form which yields ...
-1
votes
2answers
52 views

How do I find lowest upper bound and greatest lower bound when dealing with functions? [closed]

Here is my problem: I have to find the integer that is the highest lower bound for the roots of $$f(x)=x^4-3x^2+2x-4$$ I am not sure how to do this and the book I am using does not explain it very ...
0
votes
1answer
12 views

Offsetting a 2-D polynomial

I have a surface that is defined using a two dimensional polynomial: $$z = f(x) + g(y)$$ I want to offset the curve in the $XY$ plane from a point on the surface $\left(x_0, y_0, z_0\right)$ to a ...
0
votes
2answers
48 views

Find the number of distinct integer roots of $P^2 (x)-1$

Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$. Prove that the number of distinct integer roots of $P^2(x)-1$ is at most $d+2$.
-5
votes
2answers
83 views

How can I use left inverse to f(x)=3x format equation? [closed]

I want to solve linear equations as following. $$f(x)= 3x^3 -4x^2 +3x -7$$ $$f(x)= 2x^3 -3x^2 +2x -1$$ $$f(x)= 1x^3 -7x^2 +1x -2$$ But these seem that there is no $y$. How can I solve by using ...
4
votes
2answers
49 views

Is there any way to compute these sums quickly?

I have a sum of the following form (all numbers are positive integers): $$F(p) = \sum_{x=1}^{N} a_x x^p $$ Where $N$ and all $a_x$ terms are known/fixed constants. However I need to be able to ...
0
votes
1answer
25 views

Linear map with polynomials - Find a matrix

Let $F:P_3\to P_3$ be a linear map given by $F(p(x))=(x+1)p'(x)$ (where $p'(x)$ denotes the derivative). (i) Specify the matrix $A$ for $F$ in the basis $\{1,x,x^2,x^3\}$ (ii) Determine a basis of ...
11
votes
3answers
1k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
2
votes
4answers
340 views

Primitive polynomials

I am revising for a discrete mathematics exam and as quite stuck on this question. Show that the polynomial $f = x^2 + 2 x + 3 \in \mathbb{Z}_5[x]$ is primitive. How many monic primitive quadratic ...
2
votes
3answers
47 views

$\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$

Show that $\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$, where $z_0,...,z_{n-1}$ are the $n$-th roots of unity. For $k=n$ it ...
1
vote
5answers
53 views

Why is discriminant less than zero?

the question is find the range of values of $c$ for which the expression $4x^2-4x+4c^2-8$ is non-negative for all real values of $x$. I got the discriminant but I don't understand why it has to be ...
3
votes
2answers
186 views

Quadratic equation / why does $x(x-2)=0$ imply $x = 0 \lor x = 2$?

I feel silly asking such elementary questions, but hopefully this is appropriate for math.stackexchange. I'm studying to take calculus next semester but I haven't done any math in a long time, so ...
4
votes
1answer
916 views

Runge-Kutta 4 - solving system of 6 differential equations (BVP)

I'm facing a tricky problem. I need to solve a system of 6 differential equations numerically, but I don't have 6 IVP (initial value problem) conditions, instead I have 6 BVP (boundary valye problem) ...
5
votes
2answers
44 views

Determining the minimal polynomial over $\Bbb{Q}$

I was working on a homework assignment from Hungerford: Find the minimal polynomial of the element $\sqrt{1+\sqrt{5}}$ over $\Bbb{Q}$. Naturally the solution would be the polynomial with roots ...
0
votes
1answer
38 views

What are some applications of “separable” spaces?

A separable space is a space that contains a countable dense subset. For example, the space of continuous functions $C[a,b]$ is separable. Are there some practical applications arising out of this ...
0
votes
1answer
45 views

Find out whether a polynomial is irreducible or not

Let $f=X^7-(7-6i)X^3+5X^2+3+6i\in\mathbb{Z}[i][X]$. Check whether $f$ is irreducible: over $\mathbb{Z}[i]$ over $\mathbb{Q}(i)$ Probably I will have to use Einstein criterion with some ...
0
votes
0answers
12 views

does this have a unit modulus solution?

Let $z_m, m=0,1,2..N-1$ be arbitrary complex numbers and form the polynomial product in the variable $\rho$: $P(\rho) = \left [\sum_{m=0}^{N-1} (m \bar z_m \rho^m) \right ] \left ...
0
votes
2answers
48 views

Solving for the roots of a polynomial

Suppose we have a polynomial of the form: $$-x^3+3x^2+9x-27=0$$ Is there an easy way to find the solutions of $x$? I know that they will be factors of $27$, so I begin by factoring $27$ into ...
1
vote
0answers
20 views

Proof of determinant formula and coprime polynomial

Problem: Let $p(z)=p_o+p_1z+...+p_{n-1}z^{n-1}$ be a polynomial of maximum degree $n-1$. Show that $p(z)$ and $z^n-1$ are coprime if and only if $$\begin{vmatrix} p_0 & p_{n-1} & ... & ...
1
vote
0answers
18 views

Primitive elements of polynomial

Problem: Given a primitive polynomial $f(x)$ over $\mathbb F_2$ of degree 248, a) is $g(x) = x^{17}$ a primitive element? Why? b) is $h(x) = x^{23}$ a primitive element? Why? Progress: I'm ...
1
vote
2answers
70 views

How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
0
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0answers
21 views

Explanation of symmetric sum in a solution

Can someone explain me why $x+y=5$ in $\text{E8}$ clearly.
3
votes
3answers
268 views

Algebra question from practice GRE exam

The following is a question from the GRE exam GR9367: Let $n > 1$ be an integer. Which of the following conditions guarantee that the equation $x^n = \sum_{i=0}^{n-1} a_ix^i$ has at least one root ...
2
votes
1answer
9 views

Partial Derivative of the root of a polynomial with respect to its coefficients

Suppose that I have a polynomial $p(x) = c_0+c_1x^{-1}+\ldots c_{N-1} x^{-(N-1)}$ with roots $x_0,\ldots,x_{N-1}$, how do I derive an expression for the partial derivative of the root with respect to ...
1
vote
0answers
35 views

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $?

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $? In the other words, is there a way to write $a_i$ in terms of ...
6
votes
0answers
144 views

Why is this polynomial non-negative? [closed]

Show that this polynomial is non-negative: $$f(x,y)=x^2(x^2-1)^2+y^2(y^2-1)^2-(x^2-1)(y^2-1)(x^2+y^2-1)\ge 0,\forall x,y\in R$$
1
vote
2answers
28 views

Factoring a polynomial with complex coefficients

Given $$3z^2+6z+3i=0$$ Find the complex roots and write in the form $a+bi$. I want to see how to factor it when there is an $i$ being multiplied by the constant.
2
votes
1answer
99 views

Quartic polynomial taking infinitely many square rational values?

I was wondering whether the value of $$P(x)=x^4-6x^3+9x^2-3x,$$ is a rational square for infinitely many rational values of $x$. Is there a general method to check this for a polynomial (in one ...
2
votes
0answers
27 views

Intuition/Derivation for Newton's Sums?

I often come across problems in which I want to find the sum of the $k$'th powers of the roots of a polynomial. I have heard of a method known as the Newton-Girardae formulae. However, I cannot ...
3
votes
0answers
60 views

Finding exact roots

I know of the rational root theorem to find all rational zeros and Newtons method of approximating zeros, but what if all the solutions are irrational/imaginary and you need exact answers for the ...
0
votes
1answer
54 views

Roots of cubic equation

If$\frac{1+\alpha}{1-\alpha},\frac{1+\beta}{1-\beta},\frac{1+\gamma}{1-\gamma}$ are the roots of the cubic equation $f(x)=0$ where $\alpha,\beta,\gamma$ are the real roots of the cubic equation ...
2
votes
2answers
67 views

For what values of $ a, b$ does the equation have real roots?

For what values of $a,b$ does the equation $${ x }^{ 2 }+2\left( 1+a \right) x+\left( 3{ a }^{ 2 }+4ab+4{ b }^{ 2 }+2 \right) = 0$$ have real roots? For it to have real roots, the ...
6
votes
1answer
78 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
5
votes
4answers
400 views

Find a polynomial from an equality

Find all polynomials for which What I have done so far: for $x=8$ we get $p(8)=0$ for $x=1$ we get $p(2)=0$ So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$ This is where I get stuck. How do I ...
0
votes
1answer
47 views

Solving algebraic equations with radicals

I have several problems requiring assistance. Solve for x: $x\left( x-\sqrt { 3 } \right) \left( x+1 \right) +3-\sqrt { 3 } \quad =\quad 0$ I've followed the suggestion to get x^2 - (√3 -1)x + ...
0
votes
2answers
103 views

Finding roots of $2x^3-5x^2+18x+45$

solve $2x^3-5x^2+18x+45$ not exactly sure where to start on finding the zeros complex or real. There is one real zero and two complex I know that from graphing just cannot do it on paper to understand ...
1
vote
0answers
36 views

How to scale polynomial coefficients for root-finding algorithms?

I've implemented the Jenkins Traub algorithm in c++ (Github repo). While the majority of the solutions work well, it seems that a small portion of the roots are unstable. Here is a link to a ...
0
votes
0answers
42 views

Eigenvalues of a matrix formed by derivatives of a polynomial

Let $f=f(x,y)$ be a polynomial with $x\geq 0,\ -2x\leq y\leq -x/2$ (so when $x=1, -2\leq y\leq-1/2$). Denote \begin{equation*} \begin{split} f_1=&\frac{\partial \ln f}{\partial x}(1,y),\\ ...