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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An efficient way to check whether a polynomial (under certain condition) is absolutely equal to zero or not

We have a function $f$ of $N$ variables which is the product of $M$ polynomials: $$f(x_1,x_2,\ldots, x_N) = P_1 \cdot P_2 \cdots P_M.$$ Each $P_i$ is a polynomial of at most three variables ...
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1answer
124 views

Is there a specific name for this notion of extensions of fields?

As this question suggests, I quite like the notion of permuting the coefficients of polynomials. And, moreover, I have another question on this direction:If L|F is a finite normal field extension, ...
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1answer
302 views

Polynomial Long Division explanation

The wikipedia example of Polynomial Long Division starts with: Divide the first term of the numerator by the highest term of the denominator if the denominator is $x-3$, and we don't know $x$, ...
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2answers
271 views

Factorize polynomial

I am trying to factorize $-6x^5+15x^4-30x^2+30x-13$ for hours:( Could someone help me? I tried making a system of equations from $(Ax^3 + Bx^2 + Cx + D) (Ex^2 + Fx + G)$ but it is a nightmare:( In ...
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1answer
710 views

Find coefficients of a polynomial given several points on its plot

For a polynomial of order n with unknown coefficients, what are the ways to find the coefficients from n+1 points on its plot? I remember one way is to construct a fractional for each point, and the ...
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3answers
240 views

Which field will contain all the roots of a polynomial over $GF(p)$

Given a a polynomial with coefficients in $GF(p)$ and degree $d$, will that polynomial always have $d$ roots in $GF(p^d)$? The text I am reading seems to be implying that this is true but I can't see ...
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1answer
102 views

Finding polynomial roots by attempts

I barely remember that I could identify them by attempts. I would get the coefficient, see what numbers can divide it and put them into the Ruffini's rule. But I don't even know what such method is ...
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5answers
297 views

How to “Re-write completing the square”: $x^2+x+1$

The exercise asks to "Re-write completing the square": $$x^2+x+1$$ The answer is: $$(x+\frac{1}{2})^2+\frac{3}{4}$$ I don't even understand what it means with "Re-write completing the square".. ...
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345 views

Factoring a complicated 20-term polynomial

I'm trying to factorise a 20-term polynomial of the 4th degree with four variables. Ideally, I'd like to do it by hand, but I get the idea that this is pretty improbable. Can anyone point me to some ...
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1answer
277 views

Polynomial problem

From http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext2_05.pdf: Suppose that $a$ and b are positive real numbers, and let $f(x)=\frac{a+b+x}{3(abx)^{\frac13}}$ for $x ...
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1answer
105 views

What's next step to simplify this?

The fraction: $$\frac{y^2-x^2}{x-y}$$ should simplify to the answer: $$-(x+y)$$ but the best I could do was expand it to: $$\frac{(y+x)(y-x)}{x-y}$$ What's next step?
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How to simplify this to the given answer?

I'm trying to simplify: $$\frac{x^2}{x^2-4}-\frac{x+1}{x+2}$$ but I can't get to the answer: $$\frac{1}{x-2}$$ How to do it?
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5answers
437 views

How to know that $a^3+b^3 = (a+b)(a^2-ab+b^2)$

Is there a way of go from $a^3+b^3$ to $(a+b)(a^2-ab+b^2)$ other than know the property by heart?
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1answer
232 views

Factoring the polynomial $2x^2 - 2x + 2$

I saw in my book that $2x^2 - 2x + 2$ factored became $2(x^2 - x + 1)$. Why it does not became $2(x(x - 1) + 1)$? Is it wrong or correct as well?
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1answer
552 views

Finding number of roots of a polynomial in the unit disk

I would like to know how to find the number of (complex) roots of the polynomal $f(z) = z^4+3z^2+z+1$ inside the unit disk. The usual way to solve such a problem, via Rouché's theorem does not work, ...
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2answers
163 views

The Polynomial concept needs to include both variables and contants?

As in Wikipedia: In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants. So, it's only considered a polynomial if ...
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1answer
213 views

Finding the minimum value of a quadratic within a range

Given any quadratic equation of the form $y=ax^2+bx+c$, I want to find the minimum value for a specific range of $x$. My programmer brain can do it in a branchy, algorithmic way as follows, but is ...
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why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$? I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward. ...
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2answers
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How does one solve a cubic polynomial with complex coefficients?

I have found formulas online for the the roots of a cubic polynomial with real coefficients, but they all said this they would not work for polynomials with complex coefficients. I need to solve the ...
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2answers
433 views

Construct an isomorphism between fields

The first part of my question asked : State all the irreducible Polynomials in $\mathbb{Z}_2[x]$ of order 3. I was able to do this and get the following polynomials : $x^3 + x^2 + x + 1 ...
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4answers
2k views

Factoring polynomials step by step

I haven't taken a math class in a few years and I have come to realize, as I am taking one again, that I do not know/remember how to factor equations. One example from my textbook is: ...
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2answers
123 views

decompose some polynomials

[ In first, I say "I'm sorry!", because I am not a Englishman and I don't know your language terms very well. ] OK, I have some polynomials (like $a^2 +2ab +b^2$ ). And I can't decompress these (for ...
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3answers
138 views

Proofs for polynomial divisons

I've got a couple of theorems that I must prove, but I'm completely stumped. Could someone help me out? Prove: If P(x) can be divided by Q(x), then it can be divided by cQ(x), where c - any non zero ...
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1answer
92 views

Certain polynomials monotone on [-1:1]?

I'm in the process of reading a paper and I believe there's a mistake, but it could also be me not noticing something. Here's the deal: Let $\bar\Pi_k$ be the set of all polynomials of degree at ...
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0answers
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On the continuation of a polynomial

This exrcise is from the first section of Marden: Exercise 12. Let the interior of a piecewise regular curve $C$ contain the origin $\cal O$ and be star-shaped with respect to $\cal O$. If the ...
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1answer
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How to calculate complex roots of a polynomial

Calculate all complex roots of the polynomial: $8t^{4} -20t^{3} -10t^{2}-5t-3$. So thanks to matlab, I can easily find out that the roots are $t = 3, -0.5, \pm 0.5i$. Unfortunately, achieving this ...
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1answer
67 views

Asociated polynomials

Hi I have another problem..Two polynomials a(x) and b(x) are asociated iff a(x)|b(x) and b(x)|a(x)….Right? And now my problem..And polynomials are indivisible when gcd is asociated with 1..And there ...
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1answer
218 views

Trouble with a problem involving Rouché's Theorem

The problem is from Marden, the first section: The polynomial $g(z) = z^n + b_1 z^{n-1} + ... + b_n$ has at least $m+1$ zeros in an arbitrary neighborhood of a point $z = c$ if $|g^{(k)}(c)| \leq ...
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3answers
406 views

Newton's method in an elementary proof

I'm starting to learn proof and logic, all my knowledge in mathematics is: basic discrete boolean logic, introductory logic, basic Calculus, Introductory Linear Algebra. I've been trying to use what ...
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478 views

Expressing a root of a polynomial as a rational function of another root

Is there an easy way to tell how many roots $f(x)$ has in $\Bbb{Q}[x]/(f)$ given the coefficients of the polynomial $f$ in $\Bbb{Q}[x]$? Is there an easy way to find the roots as rational ...
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1answer
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Dividing in $\mathbb{Z}_m[x]/p(x)$

how can I divide for example $\frac{x^2+1}{2x+1}$ in $\frac{\mathbb{Z}_3[x]}{x^3+1}$? It's like normal polynomial dividing but here I got in first step $\frac{x}{2} (\frac{x^2}{2x}=\frac{x}{2}$).What ...
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1answer
122 views

Lower Bound of a Coefficient

Fix $\epsilon >0$. Suppose $p_{n}(z) = z^{n}+a_{n-1}z^{n-1}+ \cdots + a_{0} \in \mathbb{Z}[x]$ is irreducible and has all positive real roots. Show that independently of $n$ except for finitely ...
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1answer
831 views

Why not write the solutions of a cubic this way?

For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as: $$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + ...
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1answer
148 views

How to factorize $4x^4+12x^{10/3} y^{2/3}+ \dots $?

Does anyone know how to factorize the following expression: $$4x^4+12x^{10/3} y^{2/3}+33x^{8/3} y^{4/3}+46x^2 y^2+33x^{4/3} y^{8/3}+12x^{2/3} y^{10/3}+4 y^4$$ ?
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Different notations for roots?

I have to explain the problem I am thinking first: The solutions of $x^3+6x-20=0$ are $x = 2$, $-1+3i$ and $-1-3i$. The cubic formula for these solutions is: $$x = \sqrt[3]{10 + \sqrt{108}} + ...
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2answers
354 views

Meaning of $\mathbb{R}[x]$

I ran into this expression in a paper I was reading, and I'm confused about part of the meaning. Here $u$ and $v$ are two polynomials. $$u, v \in \mathbb{R}[x]$$ I'm not really familiar with usage ...
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461 views

An Eisenstein-like irreducibility criterion

I could use some help with proving the following irreducibility criterion. (It came up in class and got me interested.) Let p be a prime. For an integer $n = p^k n_0$, where p doesn't divide $n_0$, ...
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Finding irreducible polynomials over GF(2) with the fewest terms

I'm investigating an idea in cryptography that requires irreducible polynomials with coefficients of either 0 or 1 (e.g. over GF[2]). Essentially I am mapping bytes to polynomials. For this reason, ...
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274 views

Number of terms in a monomial symmetric polynomial

Is there a closed form expression for the number of terms in a monomial symmetric polynomial in a given number of variables for a particular partition of exponents, in terms of which/how many ...
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1k views

symmetric polynomials and the Newton identities

I want to write $P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$ in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the ...
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692 views

Is it possible to convert a polynomial into a recurrence relation? If so, how?

I have been trying to do this for quite a while, but generally speaking the partially relevant information I could find on the internet only dealt with the question: "How does on convert a recurrence ...
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Solving a peculiar system of equations

I have the following system of equations where the $m$'s are known but $a, b, c, x, y, z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply ...
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What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$?

Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value ...
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density of roots of a family of polynomials: $(1-x^2)^{v+n}$

My research has brought me to the following, very general problem. Given a fixed, but arbitrary, natural number, $\displaystyle v$, consider the following family of polynomials: The $\displaystyle ...
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Finding roots of the fourth degree polynomial: $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$.

My son is taking algebra and I'm a little rusty. Not using a calculator or the internet, how would you find the roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. Please list step by step. Thanks, Brian
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Are translations of a polynomial linearly independent?

I've been wondering about the following question: Suppose that $P$ is a polynomial of degree $n$ with complex coefficients. Assume that $a_0, a_1, \dots, a_n \in \mathbb{C}$ are distinct. Are the ...
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1answer
233 views

Factoring a trivariate polynomial

I would appreciate some help with factoring a trivariate polynomial. The polynomial in question is $$p(x,y,z)=a_1 x^7+a_2 x^5y+a_3 x^3y^2+a_4 xy^3+a_5 x^4z+a_6 x^2yz+a_7 y^2z+a_8 xz^2,$$ where the ...
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Khayyam's work on cubic equations

Omar Khayyam is known for his significant progress in solving cubic polynomial equations. For example, his biography on www-history.mcs.st-andrews.ac.uk says (...) This problem in turn led Khayyam ...
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Why are cubics separable over fields that are not characteristic 2 or 3

Why is it that cubics are separable over fields that are not of characteristic $2$ or $3$? This is the starting point for some a discussion of the Galois group of a cubic, but I seem to be stuck ...
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How to decompose function like x^2+x-6?

I was trying to wrap my brain around this but could not think of anything. I am very poor at maths but trying to learn of it. Edit: Exact question is If x^2+x-6 is a composite function f(g(x)); then ...