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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Identifying some quotient rings

How come that $k[w,z]/(w^2+z,w^3 z^2)\cong k[w]/(w^7)$? Also why is $(xz,w)=(x,w)\cap(z,w)$ in the polynomial ring in 3 variables? what are the rules of ideal calculus making these results evident?
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Finding the maximum and minimum of $V(t) = 2t^2 − 16 t + 40$

The volume of water in a tank, V m3, over a 10 month period is given by the function $V(t) = 2t^2 − 16 t + 40,$ where t is in months and $ t ∈ [0, 10].$ I completed the sure and got $2(t - 4)^2 + 24$ ...
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$f(x) = 5 + 6x − 3x^2, \ x \in [−5, 3)$

$$f(x) = 5 + 6x − 3x^2,\ x \in [−5, 3)$$ Sketch the graph of each of the functions below and state the domain and the range of each function. I found the $y$ intercept which is $5$. But the $x$ ...
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1answer
202 views

Factorizing $(x-1)(x-3)(x-5)(x-7)-64$

We need to factorize: $$(x-1)(x-3)(x-5)(x-7)-64$$ We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$. So this means that if the ...
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Show that a set of polynomials make a linear space.

I have a problem that states: "Let P be the set of all polynomials of degree at most 2. Show that P is a linear space." I know how to show that a set of vectors make a linear space with a certain ...
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2answers
163 views

Can there be parameter redundancy in an ARMA process where constant is non null?

I am asked to identify an ARMA(p, q) process from an equation, and to avoid parameter redundancy. The equation is of the form : $\varphi(L)y_t = c + \theta(L)\varepsilon_t$ with $\varphi(L) = 1 - ...
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4answers
477 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
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1answer
273 views

Automorphisms and Splitting Fields

Note: This question comes from a non-examined question sheet from an undergrad maths course. I want to find the splitting fields of the following polynomials: $x^3-1$ over $\mathbb{Q}$ $x^3-2$ over ...
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4answers
208 views

Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$

Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$ Attemption $$ \begin{cases} f(-1) = -a - b - 16\\ f(2) = 8a + 2b - 28 \end{cases} $$ But then when I plug ...
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Find the values in this polynomial

Could I please have help with this polynomial. If (2x-3) and (x+2) are factors of 2x$^3$ + a$x^2$ +bx + 30, find the values of a and b. So x = 3/2 or x = -2 When x = -3/2 = 9/4a + 3/2b + 147/4 ...
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Find the value of a and b if $( x + 1 )$ and $( x -2 )$ are factors of $ ax^3 - 4x^2 + bx - 12$ [duplicate]

Find the value of $a$ and $b$ if $( x + 1 )$ and $( x -2 )$ are factors of $ax^3 - 4x^2 + bx - 12$. This is regarding polynomials. Thank you
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1answer
69 views

Can this quartic equation be reformulated as $x =$ an expression?

$c = \sqrt{a^2 + x^2} + \sqrt{(b - x)^2 + (a - x)^2}$ Reformulated as a quartic equation: $x^4 + (-a - b)x^3 + (a^2 + ab + 2b^2 - c^2)x^2 + (-ab^2 - b^3)x + (-a^2c^2 + b^4 + c^4) = 0$ Is there an ...
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1answer
83 views

Monic polynomial reducible in rationals

Let $P(x)\in \mathbb{Z}[x], Q(x),R(x)\in \mathbb{Q}[x]$, and all three polynomials are monic. Suppose $P(x)=Q(x)R(x)$. Is it true that $Q(x),R(x)\in\mathbb{Z}[x]$? Gauss's Lemma says that since ...
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1answer
44 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
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5answers
300 views

Given $x+y$ and $x\cdot y$, what is $x^3+ y^3$ ?

I have been looking at an assortment of high school number sense tests and I noticed a reoccurring problem that states what x+y is and what $x\cdot y$ is then asks for $x^3+ y^3$. I want to know how ...
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2answers
103 views

Polynomials through successive differences

Let $h_0:\Bbb{N}\rightarrow\Bbb{N}$ be any function. Define recursively, for $m>0$, $$h_{r+1}(m)=h_r(m)-h_r(m-1).$$ Suppose that for some $k>0$ we have $h_k(m)\equiv d$ constant. Is this ...
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1answer
30 views

Proof that complex conjugate of polynomial result equals pynomial result with complex conjugated argument

This question feels uneasy to be expressed by words for me, however, I'm asked to prove this: $$P(\overline{a+bi}) = \overline{P(a+bi)}$$ Of course, $\overline{a+bi} = a-bi$.
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1answer
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Help with Spivak Calculus Ch3 Problem 6a

Yet again I find myself stuck on a Spivak question. This time it is simply the question that isn't clear to me. It states: If $x_1, ..., x_n$ are distinct numbers, find a polynomial function $f_i$, ...
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2answers
325 views

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I'm trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} ...
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All monic polynomials of degree $d$ such that $f(x) | f(x^n) \forall n \in \mathbb{Z}^+$?

The coefficients may be complex. I was doing a problem for $d=4$ and am wondering if this can this problem be generalized for any $d$
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0answers
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Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
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3answers
190 views

$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$

I am trying to show that $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$ where $n,m$ are positive integers. I showed that if $\gcd(n,m)$ is not $1$, then $X^n-Y^m$ is reducible. How to ...
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0answers
181 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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2answers
93 views

Polynomial such that $P(\sin x)=a\cos x+b$

Given four real numbers $a,b,\alpha,\beta$ with $ a\ne0, \alpha<\beta$. Does there exist a real coefficient polynomial $P(x)$ such that $$P(\sin x)=a\cos x+b$$ hold for all $x\in ...
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6answers
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How to factorize $2x^2+5x+3$?

I'm doing pre-calculus course at coursera.org and I'm in trouble with this solution $$2x^2 +5x +3 = (2x+3)(x+1)$$ By trial, using ac-method I got stuck: $$ ac = (2)(3) = 6\\ 6 + ? = 5 \Rightarrow~ ? ...
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1answer
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How to find charpoly from eigenvalues and CH to prove an equation

For an uknown 3x3 matrix $A$ we know that $\operatorname{tr} A = 0$, $\det(A) = 1/4$ and we also know that two eigenvalues are the same. Proove that $4A^3 = -3A - I$. Problem says to use Vieta to find ...
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1answer
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Factorize real polynomials to quadratic factors. Proof without fundamental theorem of algebra.

I've shown that if $P(x) \in \mathbb{R}[X]$, then exist $Q_1(X), \dotsc, Q_k(X) \in \mathbb{R}[X]$ so that $P(X) = Q_1(X) \cdots Q_k(X)$ with $\deg Q_i \leq 2$ for all $1 \leq i \leq n$. My proof - ...
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0answers
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Rational Non-Integral Root

Prove by contradiction that the following equation with integral coefficients can not have a rational but non integral root. $x^{n}+p_{n-1}x^{n-1}+p_{n-2}x^{n-2}+\cdots +p_{0}=0$
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1answer
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$P(-2)=P(-5)=n$

Prove that if $n$ is a positive integer, there exists only one polynomial $\displaystyle P(x)=\sum_{i=0}^n a_ix^i$ degree $n$ that satisfies: $(i):\,a_i\in\{0,1,\ldots,9\}$ $(ii):P(-2)=P(-5)=n$
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Show quartic polynomial has no real solutions

To show a lower bound for the runtime of an algorithm, I need to show that $$ 3 x^4 - \frac{64}{5} x^3 + \frac{192}{5} x^2 - \frac{192}{5} x+ 12 > 0 $$ for all real numbers $x\in \mathbb{R}$. ...
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1answer
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Binomial coefficient difference

I have the following difference of binomial coefficients: $$f(m)={m+n\choose n}-{m-d+n\choose n}$$ I believe the following two things should hold true: For $m$ large enough, $f(m)$ is a polynomial ...
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1answer
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Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...
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2answers
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Is this a theorem regarding the solutions of polynomials?

I wanted to refer to this, but I can't remember if this a theorem, named or otherwise, and if it is, how to properly state it. The idea is if we have a solution in radicals to a polynomial with ...
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1answer
87 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
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1answer
112 views

Is this polynomial solvable by radicals?

The polynomial $p(x) = x^6-9x^4-4x^3+27x^2-36x-23$. has at least one (real, irrational) root that is expressible by radicals (can you find it?). Are all the roots of $p$ expressible by radicals and ...
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Polynomial of degree 4 with real coefficients, two complex roots given.m

Write in the form f(z) = 0, where f(z) is a polynomial of degree 4 with real coefficients, the equation having (3 + i) and (1 + 3i) as two of its roots. Can anyone help me? I'm guessing the two ...
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1answer
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How do I determine between positive and negative inflection

Is it possible to identify whether an inflection point such as this example, contained in y = x^3 from the wikipedia: Is positive or negatively oriented (i.e. the ...
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Why $S_4$ has no transitive subgroups of order 6?

I know that every transitive subgroup of $S_4$ have to be order divisible by 4, but i should solve this with Galois Theory. I think this theorem can be usefull: Theorem 4.2. Let K be afield and f in ...
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Is it possible to find out $x^2$ parabola and function from 3 given points?

I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple $y = ax^2$ parabola to draw the falling ball. I have given two points, the ...
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How do I see that $x^5+x-1=(x^2-x+1)(x^3+x^2-1)$

I've recently been asked my friend to find the solutions to the expression $x^5+x=1$, now I haven't yet done complex analysis, but I thought I'd give it a go. I came up with a pretty, but probably ...
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2answers
130 views

Finding a least common multiple (LCM)

My Algebra 2 book explains how to find a least common multiple: Find the least common multiple of $4x^2 - 16$ and $6x^2 - 24x + 24$. Solution Step 1 Factor each polynomial. Write ...
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complex numbers, complex roots of equation.

$z_1=a+bi$ , $a,b\in\Bbb R$, $b\neq 0$ is a complex root of the equation $z^2-2z+25=0$. Without evaluating the roots, answer the following questions: i) show that $\overline{z_1}$, the conjugate of ...
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A question about degree of a polynomial

Let $R$ be a commutative ring with identity $1 \in R$, let $R[x]$ be the ring of polynomials with coefficients in $R$, and let the polynomial $f(x)$ be invertible in $R[x]$. If $R$ is an integral ...
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1answer
74 views

How to determine the existence of points on a circle (with polynomials)

Let $P\in\mathbb{C}[X]$ and $a\in\mathbb{C}$ such as $P(a)\neq 0$. Assume that $a$ is root of order $k$ of $P-P(a)$. Show, for $\rho>0$ small enough, there exist, on a circle centred at $a$ and ...
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MATLAB: Approximate tomorrow's temperature with 2nd, 3rd and 4th polynomial using the Least Squares method.

The following is Exercise 3 of a Numerical Analysis task I have to do as part of my university course on the subject. Find an approximation of tomorrow's temperature based on the last 23 values ...
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4answers
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Number of solutions of $P(x)=e^{ax}$ if $P$ is a polynomial

In MSE question the equation $x^2-1=2^x$ is considered, this is a generalization: Let $P_n(x)$ a polynomial of degree $n > 0$. It is well know that the equation $P_n(x)=0\;$ has at most $n$ real ...
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2answers
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Number of monic irreducible polynomials of degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ can exist over $F$? Thanks!
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2answers
142 views

Why doesn't the polynomial factor theorem hold for polynomials in a non-field ring?

I was reading in a book that the Factor Theorem only holds over fields (not rings). Why would that be true? No where in the proof of the factor theorem is a multiplicative inverse taken, so the proof ...
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1answer
36 views

Proving that $|\Phi_n(x)| > x-1$

Let $\Phi_n$ be the n-th cyclotomic polynomial. I'd like to prove that $$\forall n \geq 2, \forall x \in [2, \infty[, |\Phi_n(x)| > x-1$$ The result is clear when $n$ is prime, but I'm struggling ...
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1answer
271 views

Accuracy of the Newton-Cotes formulas for polynomials of degree $n+1$ and even $n$

Let $f$ be a polynomial of degree $n+1$. The Newton-Cotes formula is given by $$\int_{-t}^tf(x)\text{ d}x\approx\sum_{k=0}^nf(x_k)\int_{-t}^t\omega_{n+1}(x)\text{ d}x \tag{*}$$ where ...