This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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3answers
41 views

How can I show the uniqueness of homomorphism?

Let $R$ be a commutative ring and let $k(x)$ be a fixed polynomial in $R[x]$. Prove that there exists a unique homomorphism $\varphi:R[x]\rightarrow R[x]$ such that $\varphi(r)=r\;\mathrm{for\; ...
0
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1answer
43 views

Can a polynomial of degree $4$ have no turning points or no inflection points?

Can a polynomial of degree $4$ have no turning points or no inflection points? If yes what is an example of polynomial of degree $4$ that show these feature. If no what is the minimum number of ...
1
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1answer
46 views

Solve $(x^2+1)^2=4x(1-x^2)$

Let $(x^2+1)^2=4x(1-x^2)$. I haven't tried anything real yet, except to expand. I know it is easy but I don't have any idea for the moment. So please help!
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0answers
24 views

Proving that for every $\lambda$ there exists an $x \in \mathbb R$ so that $x(p(x)-2)^2=\lambda$, where p(x) is a non-constant polynomial.

The question is: Prove that for every $\lambda$ exists an $x$ such that $x(p(x)-2)^2=\lambda$, where $p(x)$ is a non-constant polynomial, and $\lambda \in \mathbb R$ So I've gone ahead and opened up ...
1
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0answers
45 views

To find root of $x^n+1=0$ [duplicate]

If $ \alpha_1,\alpha_n, ....\alpha_n $ be the roots of the equation $x^n+1=0$, then $(1-\alpha_1)(1-\alpha_2)...(1-\alpha_n)$ is equal to a) 1. b) 0 c)n d)2 when I put n=3,and directly evalutae ...
1
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1answer
31 views

finding root of an equation with real coefficient.

If the equation $x^4 + ax^3 + bx^2 + cx+ 1=0 $ (where a,b,c are real numbers) has no real roots and if at least one root is of modulus one, then a)b=c b)a=c c)a=b d)none of the above
-4
votes
1answer
41 views

How to prove that the dimension of the set of all the Polynomials up to degree n with real coefficient is $n+1$? [closed]

Let $P_n(R) = 1 + x + ... + x^n $ How do I prove that the dimension of $P$ is $n+1$?
2
votes
2answers
75 views

If $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the roots of the equation $x^n+1$

then $(1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_n)$ equals to ? I think here we need the info of whether $n$ is even or odd else how will we say whether by vieta's formula what is the value of ...
2
votes
2answers
68 views

$R$ commutative ring with unity , does polynomials with unit leading coefficients of degre s from $0$ to $n$ generate all polynomials of deg $\le n$?

Let $R$ be a commutative ring with unity , consider the polynomial ring $R[x]$ , let $\mathcal P_n:=\{f \in R[x] : f=0$ or $\deg f \le n\}$ , so $\mathcal P_n$ is a finitely generated module over $R$ ...
1
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1answer
23 views

We assume that there exists a ring homomorphism $f:k[x,y]/(\phi(x,y))\to k[t]/(t^2)$ that satisfy given conditions.

Let $k$ be a field, $r \in k$, and $\phi(x,y)=\sum a_{ij}x^iy^j\in k[x,y]$. We assume that there exists a ring homomorphism $$f:k[x,y]/(\phi(x,y))\to k[t]/(t^2)$$ satisfying: ...
2
votes
2answers
25 views

To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field

Is there any way to show that for $a,b \in F$ , the ideal $\langle x-a , y-b\rangle$ is maximal in $ F[x,y]$ , by showing that the quotient $F[x,y]/\langle x-a , y-b\rangle$ is a field ? Is the ...
4
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1answer
48 views

$a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$ implies $a(x)$, $b(x)$ constant?

If $a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$, then does it necessarily follow that $a(x)$ and $b(x)$ are constant? Edit. To clarify, $\mathbb{C}(x)$ is the field of rational ...
0
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0answers
29 views

Buchberger algorithm and ideals

I'm working on Groebner bases using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let $\mathbb{Q}[x,y,z]$ with the graded lexicographic order with $x>y>z$. For ...
0
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0answers
11 views

Can a polynomial form any one to one and continuous graph?

Hello I was wondering if it was possible to right any one to one and continuous graph as a polynomial with real co-efficient s. If this is not so why? It seems like it would possible to just write a ...
0
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0answers
21 views

irreducibility over integral domains

A polynomial $f(x)=g(x).h(x)$ over $D$ ,where $g(x)$ or $h(x)$ must be a unit in $D[x]$ and $D$ is an integral domain, then we say that $f(x)$ is irreducible polynomial over $D$. Since $\mathbb Q$ is ...
5
votes
3answers
87 views

What's the best way to compute $\frac{a^4 + b^4 + c^4}{a^2 + b^2 + c^2}$

So, my teacher gave us this to compute yesterday, and I'm completly confused on how should I proceed : $$\frac{1^4 + 2012^4 +2013^4}{1^2 + 2012^2 + 2013^2}$$ I've tried several ways, but most of ...
3
votes
0answers
44 views

If f/g is symmetric (resp homogeneous), must f and g be as well?

Suppose we have two polynomials $f$, $g$ in $k[X_1, ..., X_n]$ over some field $k$, and they have no factor in common. Suppose that $f/g$ is symmetric. Must than $f$ and $g$ also both be symmetric? ...
0
votes
1answer
23 views

Finding a matrix of a linear map with respect to Bases.

Let $f:U \rightarrow V$ be a linear map where $U$ and $V$ are finite-dimensional vector spaces. Let $U$ be the vector space of polynomials degree $3$ in variable $t$. $F:U \rightarrow \mathbb{R}$ be a ...
0
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0answers
20 views

Parameter estimation with polynomial cost function

I am working on a model that can be written in its simplest form as $$ \mathbf{d_1}=a\,\mathbf{d_2}+b\,\mathbf{d_3}+ab\,\mathbf{d_4}, $$ where $\mathbf{d_i}$ are some data columns and $a,b$ are ...
2
votes
2answers
60 views

Maximum value of the sum of absolute values of cubic polynomial coefficients $a,b,c,d$

If $p(x) = ax^3+bx^2+cx+d$ and $|p(x)|\leq 1\forall |x|\leq 1$, what is the $\max$ value of $|a|+|b|+|c|+|d|$? My try: Put $x=0$, we get $p(0)=d$, Similarly put $x=1$, we get $p(1)=a+b+c+d$, ...
0
votes
1answer
28 views

Find the matrix of the operator $f(T)$ relative to the ordered base of $V$

Let $V$ be a vector space of finite dimension over the field $\mathbb F$. $T$ is an operator on $V$: $T:V \to V$. $B$ is an ordered basis of $V$. The matrix $T$ relative to the basis $B$ is $A$. If ...
0
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0answers
40 views

$R_{a} = R[x]/(x)$ isomorphic to $R_{b} = R[x]/(x-1)$

I am looking at the following two rings: $R_{a} = R[x]/(x)$ and $R_{b} = R[x]/(x-1)$. I was told that these two rings were isomorphic, but I don't see why. Is this due to the minimal polynomials? ...
0
votes
1answer
33 views

Trying to Express A Factorial As A Polynomial

I'd like to express the following as a polynomial. $$(a-1)(a-2)(a-3) . . . (a-b)$$ where $b<a$ I'm currently working on it now, but wanted to see if anyone's already done it, or already know what ...
1
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1answer
42 views

If $K=F(K^p)$ is a finite extension and $\{a_1,\ldots,a_n\} \subset K$ linearly independent then so is $\{{a_1}^p,\ldots,{a_n}^p \}$

Suppose that $F$ is a field of characteristic $p$. Let $K/F$ be a finite extension and $K=F(K^p)$, where $K^p:= \{x^p\mid x\in K\}$. Suppose $\{a_1,\ldots,a_n\} \subset K$ is linearly independent ...
0
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0answers
19 views

Curve Conversion

I have a curve that is kind of spline. It is defined as a sequence of polynomial segments. It is defined by order, knots, and coefficients of the polynomials. How can I transform or convert this ...
5
votes
6answers
141 views

Let $f(x)=x^3-3x+1$.Find the number of distinct real roots of $f(f(f(x)))=3$

Let $f(x)=x^3-3x+1$.Find the number of distinct real roots of $f(f(f(x)))=3$ I have noticed that $f(x)=3$ has solutions $-1,-1,2$ But here how to find more roots ? in fact how to even say that they ...
1
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1answer
24 views

Finding $a, b, c$ values of a polynomial from a graph.

I was doing my homework and I am now stuck on question number 7 which is: The diagram shows the curve with the equation $y = (x + a)(x - b)^2$ where $a$ and $b$ are positive integers. (i) Write down ...
1
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7answers
65 views

How to tell whether the roots are the only rational roots for a given polynomial

Find all the rational roots of the polynomial $p(x)=2x^4-5x^3+7x^2-25x-15$. I only found $x=3, -\frac{1}{2}$. I am not sure whether there is any other rational roots. Is there a way to tell whether ...
0
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1answer
37 views

Find the roots of cubic equation.

If the function $$f(x)=x^3-9x^2+24x+c$$ has three real and distinct roots $l,m,n$, the find value of $[l]+[m]+[n]$ where $[..]$ represents greatest integer function In my book there is no ...
3
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2answers
30 views

holomorphic funktion is a polynomial

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be holomorphic. If we have $|f(z)|\leq|z|^n$ for some $n\in\mathbb{N}$ and all $z\in\mathbb{C}$, then $f$ is a polynomial. I tried to apply Liouville's theorem ...
0
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2answers
44 views

How to prove polynomial has repeated roots? [closed]

if we have a increasing function, say $f(x)$, so we can say $f'(x) \geq 0,\space \forall x\in \mathbb{R}$. We take a special case: if $f'(x)=0$ has a root $\alpha$ and $f(\alpha)=0$ this ...
2
votes
0answers
47 views

Rewriting the polynomial $x^2 + y^2 + z^2 - xy - xz - yz$?

I am wondering whether it is possible to factorize/rewrite the following with Newton's identities (or some other algorithm) where the polynomial is given by $x^2 + y^2 + z^2 - xy - xz - yz$. I am ...
0
votes
1answer
21 views

How to show if two polynomials are equal for all substituted real numbers, then all the coefficients are equal

Let $p(x)=c_0+c_1x+\ldots+c_lx^l$ and $q(x)=d_0+d_1x+\ldots+d_mx^m$ be polynomials with real coefficients. Suppose $\forall x\in\mathbb{R}$, $p(x)=q(x)$. Show that $l=m$ and that for all ...
4
votes
1answer
75 views

A contest math problem

Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$. If $\alpha $ and $\beta $ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$, then prove that $|\beta ...
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0answers
36 views

Solving This Polynomial?

I have the following polynomial $$\frac{(ar-1)(ar-2)(ar-3)(ar-4)(ar-5)}{(r-1)(r-2)(r-3)(r-4)(r-5)} = P$$ where $r$ is my variable and $P$, $a$ are just real constants. I was wondering whether or not ...
0
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1answer
74 views

If $\alpha^3 - 3\alpha^2 + 5\alpha -17 =0 $ and $\beta^3 - 3\beta^2 + 5\beta+11 =0 $ then find value of $\alpha+\beta$($\alpha,\beta$ is real number) [closed]

The curve $y = x^3 - 3x^2 + 5x $ is a strictly increasing curve. $y=x^3 - 3x^2 + 5x -17 =0 $ intersect x axis between 3 and 4. $y=x^3 - 3x^2 + 5x +11 =0 $ intersect x axis between -2 and -1. The ...
1
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1answer
16 views

Solving a polynomial equation along a set of lines numerically.

Assume that I for some reason want to solve multidimensional polynomial equations $$p(x_1,x_2,\cdots,x_k) = 0$$ or possibly (if there is no solution) $$\min_{\forall x_{.}} \{p(x_1,x_2,\cdots,x_k)\}$$ ...
2
votes
3answers
66 views

The number of distinct real roots of a polynomial of degree 4

Suppose I have a equation of a degree of 4 and I don't know a proper method of solving this type of equation (like completing the square is a proper method to solve the quadratic equation) so how or ...
0
votes
2answers
27 views

Roots of a Non-Monic Cubic Polynomial

Find all roots of $f(x)=231x^3+68x^2-9x-2$ I cannot use the cubic formula or Viete's theorem here because the polynomial is not monic. The only other way I can think of doing this is by the rational ...
0
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1answer
39 views

Solving equations using 3x3 determinants

Im trying to solve the following equations by use of determinants. I have scanned my work sheet (sorry for the mess) but i cant see where i am going wrong. The equations are at the top, following ...
1
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2answers
105 views

If $\,ax+b=cx+d,\,$ then is $\,a=c\,$ and $\,b=d$?

I am a high school student my maths teacher said that if $\,ax+b=cx+d,\,$ then is $\,a=c\,$ and $\,b=d.\,$ Can someone give me a prove of this?
0
votes
1answer
41 views

Solving for a cubic polynomial's roots using Viete's Theorem

I am asked to find the roots $f(x)=x^3-24x^2-24x-25$. However, the only thing I am aware of in regards to finding the roots of cubic polynomials is Viete's Theorem. However, this theorem requires that ...
0
votes
1answer
30 views

Cant find roots for this eq'n

Can someone explain how to factor/find roots to this polynomial: $$ s^4 + 14s^3 +45s^2 +650s + 1800 = 0 $$ its such a nightmare ive been stuck for hours, any help would be appreciated :)
1
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0answers
23 views

Characterization of Groebner Bases in terms of unicity of remainders

Let $I$ be an ideal of a polynomial ring $k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to a ...
0
votes
3answers
24 views

Factorization of a polynomial using synthetic division

Factorize: $$x^4-x^3-19x^2+49x-30$$ In the figure above, I have showed upto where I tried? Can anyone help me to complete it?
0
votes
0answers
29 views

Finding the Generalized Eigenspace

Given is the matrix, \begin{bmatrix}0&0&-2&0&0\\0&0&1&0&0\\1&1&2&0&0\\-1&-1&-2&-1&-2\\1&1&2&1&2\\\end{bmatrix} Find ...
6
votes
1answer
93 views

Is there any nice explanation of why the complex exponential function has no roots in the complex plane? [duplicate]

Here I am not looking for an explanation that uses basic properties that complex exponential function has, such as $e^{z+w}=e^ze^w$ or $e^0=1$ or any other, if this fact can be explained by using ...
0
votes
1answer
15 views

build function passing for specific points

I have to solve a problem very similar to this how-to-create-a-function-passing-through-given-points I need a function that draw a curve like the blue one in the picture here thus passing as ...
1
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0answers
28 views

Condition for polynomial to have repeated real roots.

Using Rolle's theorem find the condition that a polynomial equation $f(x)=0$ can have repeated real roots. And hence using this condition prove that $1+\frac{x}{1!}+\frac{x^2}{2!}\dots ...
0
votes
2answers
42 views

nth roots of the polynomial $x^3 =2$

I have to find the solution of the polynomial x^3 - 2 =0. Attempt: $x^3=2$ $x^3=2.1 =2(cos2k\pi+isin2k\pi)$ and $k=0,1,2$ $x=2^{1/3}(cos(\frac{2k\pi}{3})+i sin(\frac{2k\pi}{3}))$ now we will get ...