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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Determine if a given set of vectors span $\mathbb{R}[x]_{\leq2}$

I am working on a linear algebra question, which asks you to determine if the vectors $(1+x), (1-x), x, (1+x^2)$ span the vector space $V=\mathbb{R}[x]_{\leq2}$. I think that the four vectors do ...
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A cheap error estimate and a costly doubt

Carl de Boor poses the following problem in his A Practical Guide to Splines (1978 - Chapter II, p. 38, problem 4): The calculation of $||g|| = \max\{|g(x)| : a \le x \le b\}$ is a nontrivial ...
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Why is the degree condition for a degree reverse lexicographic order necessary?

A degree reverse lexicographic order $\prec$ is defined as follows: Given the polynomial ring $R=K[x_1,...,x_n]$. Two monomials in $R$ have the order $x^u\prec x^v$, if $\deg(x^u)<\deg(x^v)$, or ...
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Limits of Taylor POlynomials over $k$-tuples?

Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix $\textbf{a}$ $\in E$, and suppose $\textbf{x}$ $\in R^{n}$ is so close to $\textbf{0}$ that the points \begin{equation*} ...
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Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{1}{1-a}+\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can ...
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How to calculate the sum $\sum_{i=0}^\infty i^nx^i$ [duplicate]

Here I have $x\in\mathbb{R}_+$ and $x < 1$. I would like to evaluate the following sum: $$\sum_{i=0}^\infty i^nx^i.$$ I know that $$\sum_{i=0}^\infty x^i=\dfrac{1}{1-x}.$$ So I started ...
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What is an irreducible polynomial in $\mathbb{Z}$ that has root $\sqrt{2}+\sqrt{3}$?

What is an irreducible polynomial in $\mathbb{Z}$ that has root $\sqrt{2}+\sqrt{3}$? Obviously that root is not in $\mathbb{Z}$. I tried ...
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1answer
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Ring Homomorphism Textbook Question

Please help me understand the last three sentences in this paragraph from the Artin textbook. Where does this come from: "The monomials that appear in $r_0(t^2)$ have even degree, while those in ...
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$T(M) \cong R<x_1,\ldots,x_n>$ isomorphism question

I have that $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$ where $T^k(M) = \bigotimes_{i =1}^{k} M$. In a paper i have that to prove such isomorphism, we define: $$\Phi:R<x_1, \ldots, x_n> ...
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$f(x)f(1/x)=f(x)+f(1/x)$

Find a function $f(x)$ such that: $$f(x)f(1/x)=f(x)+f(1/x)$$ with $f(4)=65$. I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$ which leaves $f(1/x)$ as: ...
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1answer
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A quadratic polynomial bounded by another

Suppose $p(x)$ and $q(x)$ are two quadratic polynomials in real coefficients such that: $$\lvert p(x) \rvert \leq \lvert q(x) \rvert ~ ~ ~ \text{for all} ~ x \in \mathbb{R} \tag{1}$$ Is the above a ...
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1answer
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What polynomial with real coefficients generates all polynomials with real coefficients that satisfies $f(2+i)=0$?

What polynomial with real coefficients generates all polynomials with real coefficients that satisfies $f(2+i)=0$? Obviously, the polynomial $f(x)=x-2-i$ satisfies the constraint but does not have ...
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1answer
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$\left( 1 - \frac{1}{n} \right)\left( 1 - \frac{2}{n} \right) \cdot … \cdot \left( 1 - \frac{k-1}{n} \right) = \frac{n!}{n^k r! (n-k-r)!}$

I'm trying to understand a proof in "Interpolation and Approximation by Polynomials" by Phillips. Let me quote (page 253): "For $k\geq 1$ we begin with $$B_{n+k}^{(k)}(f;x)=\frac{(n+k)!}{n!} ...
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Please help solve this equation [on hold]

$$3.8r - 0.057r^2 + 0.00038r^3 + 0.00000095r^4 = 95$$ Please help.
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ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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1answer
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Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$

Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$ I simplified $x^{100} + 2x + 10$ to $x^{15} + 2x + 10$ and $x − 11$ to $x+6$ to be in $\mathbb Z_{17}$. ...
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About polynomials. If there is not two plynomials with the same grad in S, then S is linearly independent.

The problem states as follows. Let S be a set of polynomials non zero over a field F. If there is not two plynomials with the same grad in S, then S is linearly independent. I tried the following. ...
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1answer
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Ring polynomial kernel generators

This is the textbook question: Q: Find generators for the kernels of the following maps: $\mathbb{R}[x,y] \to \mathbb{R}$ defined by $f(x,y) \mapsto f(0,0)$ $\mathbb{R}[x] \to \mathbb{C}$ defined ...
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Is it possible to find the value of $x$ where $e^x$ exceeds $x^{10}$ by hand?

All I managed is to "simplify" the equation $e^x=x^{10}$ to $\frac{x}{\ln{x}}=10$. Is there some way or trick to make the equation look like $x=\dots$? (Solve the equation, in other words.)
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Polynomial and a field [on hold]

How to prove that if a polynomial $$f(x) = ax^3+bx^2 +cx +d,$$ where $a,b,c,d \in K$, where $K$ is a subfield of $\mathbb{C}$, has a root in $K(\alpha)$ then $f$ has a root in $K$. $\alpha \in ...
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1answer
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Solving an equation with complex numbers

I want to use complex numbers to solve the following problem: $x^2 = 95 - 168i$. I am sure there are a few ways of doing this but the way I want to do it is to let $x = a + bi$ and then solve for $a$ ...
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3answers
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Find the least degree Polynomial whose one of the roots is $ \cos(12^{\circ})$

Find the least degree Polynomial with Integer Coefficients whose one of the roots is $ \cos(12^{\circ})$ My Try: we know that $$\cos(5x)=\cos^5x-10\cos^3x\sin^2x+5\cos x\sin^4x$$ Putting ...
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Symmetric functions and roots of polynomials

$f = x^3-\frac{1}{2}x^2+1$ and their roots $a,b,c$. I want to find polynomial of degree 3 with roots $a^4,b^4,c^4$. I know that i need express $e_i(a^4,b^4,c^4)$ in terms of $e_i(a,b,c)$ those ...
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Historical enquiry about polynomials

I'm researching the historical use of geometry to find solutions to polynomial equations. I'd like to ask for those familiar with this topic, how would you describe the use of geometry to solve ...
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How can I maintain notes while self studying Maths?

Thank you for stopping by this thread. I'm an engineering student rekindling an interest in Maths. I just love studying Maths in my free time (and sometimes it trespasses into my non free time). I ...
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1answer
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Finding roots in finite fields.

On pg. 587 (in the finite fields chapter) of Abstract Algebra, 3rd ed. by Dummit and Foote, the following statement is made: 'If $f_1(x)=x^4+x^3+1$, $f_2(x)=x^4+x+1$ are two of the irreducible ...
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Discriminant of a polynomial modulo a prime

If $p$ is a prime and divides the discriminant of an irreducible polynomial $f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in \mathbb{Z}[x]$ why is then $disc(f(x)\bmod p)=0$? I know that the ...
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When can I divide both sides of an equation if one side is zero

Where K is some positive Integer For the following examples: $$ K(a+b)(p+q)=0 $$ $$ Ka^2+Kbx+Kc=0 $$ Can I just divide both sides of the equation by K (dividing into 0 on the right) and effectively ...
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1answer
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Show that $B^{(1)} = 0$ and $B^{(2)}$ have basis $\{[x_i,x_j]; i>j\}$

Definition: A polynomial $f \in K \langle X \rangle$ is called a proper polynomial, if it is a linear combination of products of commutators: $$f(x_1,x_2,...x_n) = \sum ...
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3answers
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$\mathbb Q [\sqrt{2} i]$ contains neither $\sqrt[4]{2}$ nor $\sqrt{2}$

I want to prove that $x^4-2$ is irreducible over $\mathbb Q [\sqrt{2} i]$. In order to verify it has no linear factors and quadratic factors, I need to show $\mathbb Q [\sqrt{2} i]$ contains neither ...
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1answer
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can I draw a 2nd degree 2 dimensional surface trough 3x3 points, like I can draw a 2nd degree polynomial through any 3 points

A Nth degree polynomial f(x) fitting N+1 points, say at regular distances like x = 1,2,3,4,5,... can be used conveniently to interpolate for values of x in between the given ones. I have a set of ...
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1answer
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Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$

Find all $ f : R\rightarrow R $ such that $f(x^2+f(y))=(f(x))^2+y, \forall\text{ x,y}\in R$ Thanks in advance!
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1answer
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Polynomial Equation Solution

Use Demoivre's theorem to show: $cos 7θ = 64 cos7 θ − 112 cos5 θ + 56 cos3 θ − 7 cos θ$ Hence,solve: $128x^7 −224x^5 +112x^3 −14x+1=0$ I've shown the first part and multiplied the equation by 2 and ...
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What sequence of polynomials is equal to $2^n$ for integers $1$ to $k$?

I am trying to prove to someone that no matter how many terms you have of a sequence you can never be 100% sure of the underlying formula. Consider this sequence: $$2^n=1,2,4,8,16,...$$ But just given ...
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1answer
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For a field $K$ is $K\subset{K[X_{1},…X_{n}]}$

Let $K$ be any field and $K[X_1,...X_n]$ the ring of polynomials in $X_1,...X_n$ with coefficients in $K$. I am wondering if $K$ is a subset of $K[X_1,...X_n]$. I believe $K\subset{K[X_1]}$ since ...
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Bounds of Zeros of a Higher Degree Polynomial Equation

The Equation is follows: $$\lambda^{k+1}+\frac{A}{A+B} \lambda^{k-l}+\frac{B}{A+B}=0$$ Where $l$ and $k$ are positive integers such that $l<k$. Also A and B are complex numbers. I would like to ...
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Interval of Polynomial Root Finding

Let's say we have a polynomial of a given degree. You don't have any tools to figure out the amount of roots in this polynomial. All you know is the function and you cannot graph it. How would you ...
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1answer
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Interval in which roots lie.

For a quadratic equation, we have several conditions from which we can determine the interval in which the roots lie. eg: If exactly one of the roots of a general quadratic equation lies in the ...
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$x^4 + 1$ reducible over $\mathbb{R}$… is this possible?

I am seeing this on a homework and am wondering if this is a typo. I am aware that $x^4 + 1$ is irreducible over $\mathbb{Q}$. I know the following: A polynomial being irreducible over some ring ...
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1answer
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Solving the following quartic equation : $x^4- {31\over 4}x^3 + {21\over 4}x + {9\over 2} = 0$

So, I don't know if it's normal but this quartic seems long to solve... When I use the ferrari method, I get the following reduced quartic : $y^4 - {2883\over 128}y^2-{27103\over ...
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2answers
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Basis for the vector space P2

I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial $ ax^2 + bx + c $ with the matrix $ A = \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \\ \end{bmatrix} ...
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Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
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Polynomial: Is there a theorem that can save my proof when $K$ doesn't include $\mathbb C$

Suppose $f(x),g(x)\in K[x]$ ($K$ a number field), let $f(x)=x^{3m}+x^{3n+1}+x^{3p+2}$, where $m,n,p\in\mathbb N$, and let $g(x)=x^2+x+1$, prove: $$g(x)\mid f(x)$$ I think this problem is not ...
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1answer
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scaling up data series based on polynomial equation

I have yearly time series data starting from 1990 to 2100 AD (x-axis). The value for 1990 is 0 and 2100 is 700, and it's increasing in each year (but not linearly). Based on this series, I come up ...
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Given $n \in \Bbb Z$, determine $\gcd(3n^2 + 7n + 4, n + 2)$.

I factored $3n^2+7n+4$ to $(3n+4)(n+1)$ and because there isn't a common factor of those and $n+2$ I said that the gcd is $1$, but is there any othere way to go about it that would come up with a gcd ...
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2answers
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Proofs involving endomorphisms on the space of polynomials

Define endomorphisms $D$ and $E$ on the space of polynomials with rational co-efficients $ \mathbb{Q}[x] $ such that $ D(x^n)= nx^{n-1}, E(x^n) = \frac{1}{n+1}x^{n+1} $ We must show that $ DE = I $ ...
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1answer
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How do we determine the decomposition of $p\mathcal{O}_K$ in $K = \mathbb{Q}(\sqrt[3]{5})$?

Let $K = \mathbb{Q}(\sqrt[3]{5})$, and $\mathcal{O}_K$ be its ring of integers. In general, how do we decide the decomposition of $p\mathcal{O}_K$, for an odd prime $p$? I know that by Kummer's ...
4
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2answers
175 views

A proof about polynomial division

Suppose $g(x)=ax+b$,$a,b\in K$,$K$ is a field, and $f(x)\in K[x]$, prove: $$g(x)|f^2(x)\Leftrightarrow g(x)|f(x)$$ The $\Leftarrow$ part is so trivial. But for the $\Rightarrow$ part I get ...
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1answer
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Unbounded polynomials

Let $p(x)$ be a polynomial of degree $d$ on $R^n$, and let $\tilde{p}(x)$ be the homogeneous components with degree $d$, then how do we prove that: if $\tilde{p}(x)$ is unbounded below, then $p(x)$ ...
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2answers
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For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies?

For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies? I need help adding rigor to my observation to create a formal proof. ...