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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10
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2answers
97 views

Let $f(x)=x^2+12x+30$. Solve $f(f(f(f(f(x)))))=0$

Here is my solve, is it correct? I figured out that we can restate $f(f(x))$ as $((x+r)(x+s)+r)((x+r)(x+s)+s)$ thus $f(f(f(f(f(x)))))=0$ is $(x+r)^2(x+s)^2(4s+3r)(4r+3s)$ from vieta's ...
0
votes
0answers
8 views

A x B^-1 has irreducible char poly when A,B random integer matrices — simple proof?

Let $A,B$ be $d\times d$ integer matrices with each entry drawn uniformly from $[0,2^n)$, and define the rational matrix $C = A \times B^{-1}$. Is there a simple way to prove that $C$'s characteristic ...
1
vote
2answers
27 views

Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function?

Two of the roots of $f(x) = x^3+bx^2+cx+d$ are $3$ and $2+i$. How do I find b+c+d? The answer choices are -7, -5, 6, 9, and 25.
-2
votes
1answer
24 views

Number of polynomials [on hold]

I have the polynomial ring $\mathbb{Z}_3[x]$. How many polynomials $p(x)\in\mathbb{Z}_3[x]$ have degree $2$?
1
vote
1answer
35 views

Equivalent quadratic forms

Two quadratic forms $$Q(x_1, x_2, \dots , x_n) \\ \text{ and } Q'(x_1, x_2, \dots , x_n)$$ are called equivalent $$\Leftrightarrow Q'(x)=Q(Tx), \text{ where } T \in M_n(K), \text{ invertible }$$ ...
2
votes
0answers
30 views

Possible integer roots of polynomial with real coefficents

If $p\in\mathbb{Q}[X]$, then the rational root theorem gives us possible integer roots of $p$. If $p\in\mathbb{R}[X]$, the theorem cannot be applied. Nevertheless, triangular inequality gives us lower ...
0
votes
1answer
38 views

How to solve quartic polynomial equation

Can someone tell me how to solve $x^4 + 6x^2 + 5 = 0$? I know what to do when each term has an exponent one less than the previous term (e.g., $x^4 + 3x^3 + 6x^2 + 5 = 0$), but not when exponents are ...
0
votes
1answer
26 views

A quadratic form over $K-$vector space $V$

Let $K$ a field, $\operatorname{char} K \ne 2$. Definition: A quadratic form over $K$ is a homogeneous polynomial $Q(x_1, x_2, \dots , x_n) \in K[x_1, x_2, \dots , x_n]$ of degree $2$. If ...
1
vote
4answers
39 views

Field Isomorphisms and Square Free Integers

I need to prove the following: Let $D$ be a square free integer. Show that $ \lbrace\begin{pmatrix} a & bD \\ b & a \end{pmatrix} \mid a,b\in\mathbb{Q}\rbrace $ is a field ...
-1
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0answers
11 views

The Roots of Jacobi Polynomials

How can i obtain the roots of Jacobi polynomials of order n>50 ? ( α<0, β<0 and $\alpha+\beta=-1$ )
0
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1answer
19 views

Question on a fairly rigorous looking proof concerning the roots of a polynomial (resultants, symmetric polynomials, Viete)

Sorry for the big reading here. I tried to get as much on here so that it would make sense later on. Even though I put quite a bit on here, I actually just have one question about what is said ...
0
votes
0answers
8 views

Bound all $k$-th derivatives by directional derivatives of order $k$

Assume $f\in C^k(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $|(\partial_\xi)^kf(x)|\leq 1$ for all $\|\xi\|=1$. Which bounds do we have for $|\partial^\alpha f(x)|$ when $|\alpha|=k$? For example, if ...
0
votes
0answers
11 views

Rational representation of conics

Currently I'm beginning my study of rational curves (Rational Bezier and NURBS) all books that I've read tell me that is "well known" that conics can't be represented by Bezier or even a B-Spline. ...
1
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0answers
34 views

Coefficients of the polynomials generated by $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$.

Let $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$ for $i\geq0$, i.e., $f_i=\dfrac{\sqrt{f_{i+1}^2\mp4}+f_{i+1}}2$ I have observed that $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ ...
0
votes
1answer
21 views

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function? $Y$ and $X$ are real variables, $a$ and $b$ are parameters. Someone said this is a product of polynomial and exponential function. Do we ...
0
votes
2answers
30 views

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, how do I get the other root

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, then I know I can just take $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ and divide it by $x+1$ to get the other root. In a ...
0
votes
1answer
44 views

Algebraic relations between trigonometric numbers

Given $n\in2\Bbb N$, what is precise algebraic relation between $cos\frac{\pi}{n-1}$,$cos\frac{\pi}{n+1}$? Both numbers are algebraic, which implies there should be an algebraic relation between ...
4
votes
0answers
31 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
2
votes
0answers
34 views

Cleaning Up Messy Product Notation

Suppose I have the following: Let $N_1<...<N_m$. Let $T_{N_k}(x)=\sum_{i=0}^{N_k}{\frac{x^i}{i!}},$ $ t(i,j,x)=(T_{N_i}-T_{N_j})(x)$ I'm trying to define a polynomial $p_{k,m}(x)$ like ...
1
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2answers
45 views

Expression for Taylor's formula with a remainder

Assume $f$ has a continuous second derivative $f~''$ in some neighborhood of $a$.Then, for every $x$ in this neighborhood, we have $f(x) = f(a) + f~'(a)(x-a) + E_1(x)$ , where $E_1(x) = \int_a^x ...
0
votes
1answer
54 views

What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
-1
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0answers
15 views

polynomial and rational functions equation, find the values of a and b that will make the statement true

Find the value of $a$ and $b$ to make the given condition true: $ax^3-bx^2+45x+54$ has $3$ as a zero and yields a remainder $12$ when divided by $x+1$
1
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0answers
15 views

Sum of $p$th powers using polynomial interpolation

It is well known that the sum of the first $n$ $p$th-powers is polynomial in $n$ and is given by: $$ \sum_{k=1}^n k^p = \frac{1}{p+1} \sum_{j=0}^p (-1)^j {p+1 \choose j} B_j n^{p+1-j} $$ where $B_i$ ...
1
vote
1answer
24 views

I have a question about Viete's formulas

If I have a polynomial $a_n x^n + a_{n-1}x^{n-1}+ \cdots + a_1 x + a_0$, and the roots of the polynomial is $r_1,r_2,\ldots,r_n$, then I can rewrite the polynomial as, $a_n x^n + a_{n-1}x^{n-1} ...
2
votes
3answers
33 views

$x^2+3$ has two zeros over ${\Bbb F}_p$ provided that $x^2+x+1\in{\Bbb F}_p[x]$ has two?

The following is an exercise in abstract algebra: If $p=1\pmod{3}$, then $x^2+x+1\in\Bbb{F}_p[x]$ has two zeros. Prove in this case that $-3$ is a quadratic residue mod $p$. Showing that ...
1
vote
0answers
13 views

Kneser Inequality in multivariables

Based on the Kneser Inequality ("Polynomials and Polynomial Inequalities", p. 260) one has $\Vert q \Vert_{[-1, 1]} \Vert r \Vert_{[-1, 1]} \leq C(n, m) \Vert q r \Vert_{[-1, 1]}$ where all norms ...
0
votes
1answer
47 views

Polynomials, prove exercise question about question

There is a polynomial P with integer coefficients and with pairwise different integers $a,b,c$ . Prove that it is not possible for $P(a) = b$, $P(b)=c$, $P(c) = a$ First off I don't understand ...
0
votes
1answer
35 views

Factoring polynomial $x^3−2x^2−4x−8$ that fails Bezout's identity test

I usually factor 3rd degree polynomial in two steps. First, I find all the divisors of the last, coefficient-free part of the polynomial (in this case that's 8) and try (applying Bezout's identity) to ...
3
votes
1answer
51 views

Polynomial with n real roots

Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_{1}x + 1$ where $a_i$ are nonnegative and real. Assume $P$ has $n$ real roots. Prove $P(2) \geq 3^n$. I thought I had a good idea about ...
1
vote
1answer
17 views

Taylor theorem for f(x+h)

I am following a proof that applies Taylor's theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf) I am not understanding one of the terms explained on ...
2
votes
1answer
25 views

Find $a$ and $b$ such that $g$ divides $f$ evenly

$f=2X^4-3X^2+aX+b,\ g=X^2-2X+3, \ f,g \in \mathbb{Q}[X]$ I have tried to divide $f$ by $g$ but I get $ (a+10)X +b +3$ as the remainder which looks like a bad result. I have, also, tried to factor ...
0
votes
1answer
16 views

Definition of monomial

I thought the definition of a monomial is an algebraic term that has no subtraction or addition. I saw on my online college homework that 2/x is not a monomial. Why?
1
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3answers
36 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
1
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0answers
13 views

Efficient Computation of Swinnerton Dyer Polynomials

the Swinnerton-Dyer polynomials are defined as $$SD_n(x) = \prod(x \pm \sqrt{2} \pm \sqrt{3} \pm ... \pm \sqrt{p_n})$$ where the product is taken over all possible permutations of $+$ or $-$ signs. ...
0
votes
0answers
23 views

Polynomial “factorization” on a set of polynomials

Considering a set of multivariates polynomials $\{P_1(x,y,...),...,P_n(x,y,...)\}$, I wish to know if a given $P(x,y,...)$ could be expressed as a sum of products of $P_i$, such as ...
-1
votes
4answers
55 views

Can I find solutions to $a^4 + a^2 + a = b^2 + b$, $a,b \in \mathbb{Z}$ and $ 1 < a < b$?

I was wondering if anyone could point me in the correct direction for either finding a solution to my problem or proving that it does not exist. $$a^4 + a^2 + a = b^2 + b \;\text{ for }\; a,b \in ...
1
vote
1answer
29 views

How can I prove this about the tangent line formula??

The equation of a tangent line to $f(x)$ at $x = t$ is $y = f'(t)(x - t) + f(t)$. Recently, I heard that it is also determined by the remainder of polynomial division of $f(x)$ by $(x-t)^2$. For ...
0
votes
1answer
25 views

CHKMO 2015 and cubic equations

Let $a,b,c$ be distinct real numbers. If the equations $E_1: ax^3+bx+c=0, E_2: bx^3+cx+a=0$ and $E_3: cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real ...
3
votes
1answer
24 views

Polynomials with purely imaginary coefficients?

Finished a homework problem concerning polynomials with all real coefficients and why complex roots of p(z)=0 come in pairs. Curious is there is a similar situation for polynomials with all purely ...
1
vote
1answer
48 views

Does Euclidean division not work for general polynomials?

If $K$ is a field. Then in $K[X]$ there is an Euclidean algorithm and if $K$ is replaced by any arbitrary commutative ring $R$, then almost we have an Euclidean algorithm, by the following result: ...
0
votes
4answers
63 views

How many solutions has this third degree equation?

how many solutions has this equation: $$ {x}^{3}+4\,{x}^{2}-1=0 $$ i tried ruffini so far and it is not working, now i'm stuck and no idea of how to aproach this.
1
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1answer
28 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
0
votes
1answer
19 views

Does the equality $\partial^\alpha(x^\alpha)(0)=\alpha!$ hold?

Do we have $\partial^\alpha(x^\beta)(0)=\alpha!=\beta!$ if $\alpha=\beta$ and $0$ else? I tried to proof it on induction, can include my attempts if needed, but they seem to have failed anyway...
1
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2answers
40 views

polynomial of $4^\text{th}$ degree, prove

There is a polynomial $f$ of integer coefficients such that $\text{deg(f)} \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no ...
3
votes
6answers
143 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
1
vote
2answers
60 views

transformation of $y=3(4-x)^3-6$

I am looking for the expansion of $y=3(4-x)^3-6$. I got confused about the $(4-x) $ part. Please help, thanks!
1
vote
1answer
32 views

Polynomial rings, division algorithm

Let $m,n$ be non-negative integers and $m>n$. Find polynomials $g(x),r(x)$ from the ring $R[x]$ such that $x^m -1 =q(x)(x^n-1) + r(x)$ , $r(x)=0$ or $\deg(r(x))<n$. In which case $x^n -1|x^m - ...
12
votes
1answer
99 views

$P(z)=0$ iff $Q(z)=0$, $P(z)=1$ iff $Q(z)=1$. Prove that $P(x)=Q(x)$ for all $x$

Assume $P(x)$ and $Q(x)$ are polynomials with complex coefficients with degree greater than or equal to $1$ such that $P(z)=0$ if and only if $Q(z)=0$, $P(z)=1$ if and only if $Q(z)=1$. Prove that ...
0
votes
0answers
26 views

I need help to solve this problem [duplicate]

Let $m,n$ be negative integers and $m>n$. Find polinomials $g(x),r(x)$ from the ring $R[x]$ such that $x^m -1 =q(x)(x^n -1) + r(x)$ , $r(x)=0$ or $deg(r(x))<n$. In which case $x^n ...
0
votes
1answer
13 views

Do monomials' degrees always depend on the whole-number exponent of the variable or whether it's a constant (having a degree of zero)?

Is it true that the monomial $4x^4$ has a degree of $4$ because of the exponent? Also, I think $-2x$ has a degree of $1$ because it has an exponent of $1$ when it's also written like this: $-2x^1$. ...