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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4answers
93 views

Can you please help factor this polynomial?

Can someone give me a step by step tutorial on factoring this polynomial? $a^3 + 6a^2 - ab^2 - 6b^2$
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2answers
34 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
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2answers
442 views

If Horner’s rule is used what is the number of multiplications to compute P(n) for an polynomial

Let P(x)=3x^2 + 4x + 2 be a polynomial in the variable x. If Horner’s rule is used, what is the number of multiplications to compute P(10)
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2answers
75 views

Tricky Question on Induction and Characteristic Polynomials

I am to prove via induction that for any $n \times n$ matrix $A$, the characteristic polynomial of $A$ has degree $n$; $(-1)^n$ as the coefficient of the $\lambda ^n$ terms; $(-1)^{n-1}\cdot ...
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1answer
76 views

Analyzing a fourth degree polynomial

Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$ $acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots. I have never solved a ...
2
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2answers
38 views

Find the value of $k$ such that $p(x)= kx^3 + 4x^2 + 3x - 4$ and $q(x)= x^3 - 4x + k$ , leave the same remainder when divided by $(x – 3)$.

$p(x)= kx^3 + 4x^2 + 3x - 4$ and $q(x)= x^3 - 4x + k$ , leave the same remainder when divided by $(x – 3)$. (a) -1 (b) 1 (c) 2 (d) -2 I am getting the value of k: $-17/29$ after equating the ...
7
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2answers
283 views

Real roots of a polynomial

Let $p$ be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that $p$ has at least two real roots. Thanks!
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0answers
49 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
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2answers
104 views

Does $s(0) = s(1)$ define a vector subspace in $\mathbb C[X]$?

I believe that $s(0) = s(1)$ does not define a vector space in $\mathbb C[X]$, but I am unsure how to show it. I know it doesn't satisfy the zero vector condition, nor is it closed under vector ...
0
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1answer
28 views

how to use Newton's polynomials calculate this?

Say we have Fibonacci recurrence: $ F_{n+2}=F_{n+1}+F_n$, with $F_0=1,F_1=1$ We can write $F_n = a \alpha^n + b \beta ^n$, so how do we use Newton's polynomials to determine the value of $\alpha^r + ...
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2answers
58 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
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1answer
22 views

Seeing complex roots on the graph of a polynomial

When I sketch the graph for a general second degree polynomial $y = ax^2 + bx + c$ it is easy to "see" its roots by looking at the points where $y=0$. This is true also for any $n$-degree polynomial. ...
2
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2answers
61 views

Does there exist a polynomial function for every n points, whose extremas are these points?

Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points? Given 3 points: $ P_1 = (0,4), P_2 = (2,2), P_3 = (4,7) $ And ...
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2answers
72 views

Determine whether S is a subspace of P3. Vector space of all real polynomials.

ATTEMPT: Have given a small attempt just really confused on how to approach. So I got the general equation of $p(x)= a + bx +cx^2 +dx^3$. So we find the derivative? and find the values of ...
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1answer
56 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
3
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1answer
65 views

a polynomial about continuous function

Let $\{a_i(x):\mathbb{R}\rightarrow \mathbb{C}\}$ be continuous functions, does there exist some continuous functions $\{\lambda_i(x)\}$ such that $$a_{n-1}(x) y^n+a_{n-2}(x) y^{n-1}+\cdots ...
4
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0answers
78 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ ...
2
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1answer
60 views

Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
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2answers
43 views

Conditions of the system of equations.

Find m to the equation:$$\left\{ \begin{array}{l}2x^3-\left(y+2\right)x^2+xy=m\,\,(1)\\x^2+x-y=1-2m\,\,(2) \end{array} \right.$$have experience My try: From $(1)$ and $(2)\,\Rightarrow $: ...
2
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1answer
40 views

Projecting an affine hypersurface away from a point in its projective closure is never a finite map?

Let $X\subset \mathbb{A}_k^r$ be an irreducible hypersurface defined by a polynomial $g$, where $k$ is an algebraically closed field. Embed $\mathbb{A}^r\hookrightarrow\mathbb{P}^r$ in the usual way. ...
0
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1answer
51 views

If f is 0 at enough points, it is the 0 polynomial?

Let $f \in \mathbb{C}[x_1,...,x_n]$, and let d be the largest $x_i$-degree of f for $0 \leq i \leq n$. Prove that f is the zero polynomial, if $f(a_1,...,a_n)=0$ for all points $(a_1,...,a_n) \in ...
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1answer
32 views

Linear algebra, polynomial problem

Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :( Let $S$ be the following subset of the vector space $P_3$ of all real ...
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1answer
29 views

If $abx^2 = (a-b)^2 (x+1)$ then $ [1 +(4/x)+(4/x^2)]^{(1/2) }=$?

As the title says. I found this question in our next term's book. A) (a+b)/(a-b) B) (a-b)/(a+b) C) a^2 +ab D) none
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1answer
40 views

Roots for quintic equations

I have been pondering over this question for a few months now. Why exactly do quintic equations have no close general expression for their roots? Looking at graphs and reading about it hasn't really ...
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1answer
25 views

Linear algebra, question about polynoms

A,B are matrices n*n over a field F. I am given a polynom f(t) {belongs to F[t]} . How can I show that Af(BA)B= ABf(AB)? I defined a polynom g(t)= t*f(t). Then I substituted AB instead of t, but I ...
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0answers
39 views

Solving a septic equation

Solve the septic equtions \begin{eqnarray*} 0 &=&c_{2}^{2}x^{7}+c_{3}c_{2}^{2}x^{6}+\left( -c_{1}^{2}+2c_{1}c_{3}c_{4}c_{2}\right) x^{5}+\left( 2c_{3}^{2}c_{1}c_{4}c_{2}-c_{3}c_{1}^{2}\right) ...
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1answer
83 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
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0answers
24 views

Using the discriminant find the real solutions?

I have an equation: $$x^4+ax^3−b^2$$ for which the discriminant is $$−b^4(256b^2+27a^4)$$ If $$b≠0$$ what are the 2 real solutions to the equation? For these two solutions, what is a=?
2
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2answers
114 views

Help in this proof in Lang's Algebra book

I'm trying to understand this part of the proof: I didn't understand why not all coefficients of $f_2,\ldots,f_n$ can lie in the maximal ideal, maybe I'm forgetting something, it should be a very ...
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6answers
158 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
2
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1answer
34 views

Factorising a complex polynomial over C

If $f(z)=z^3+7z^2+16z+10$, find all factors of $f(z)$ over $C$. If I had at least one zero or factor I would be able to find the others, but I just don't know how to start.
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2answers
15 views

Counting polynomials which are constant on a specific hyperplane

Say a function $f:\mathbb{R}^m\to\mathbb{R}$ is having the 1-property if it equals $1$ on the hyperplane $\sum_{i=1}^m x_i=1$. How many polynomials of degree $n$ have the 1-property?
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Curve Fitting a Cyclical Pattern of Data

I'm analyzing phonological characteristics of the 22 letters used in the Hebrew alphabet, and assigned each letter an enumeration to see if they are organized based on place of articulation: ...
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2answers
54 views

Determine whether this is a subspace of $P_3$

Let $S$ be the following subset of the vector space $P_3$ of all real polynomials $p$ of degree at most 3: $$S=\{p\in P_3\mid p(1)=0, p^\prime (1)=0\}$$ where $p^\prime$ is the derivative of $p$. ...
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1answer
33 views

Legendre Polynomial Orthogonality and Size

Show $(P_i,P_j)=\begin{cases} 0& i \neq j \\ \frac{2}{2j+1} & i = j\end{cases}$ for $0 \leq i, j\leq2$ I'm just not sure exactly what I'm supposed to do. Do I plug in values of i and j and ...
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4answers
40 views

How do I work with equations with more than two variables?

I was trying to rewrite this equation in terms of $s$: $$ p = 4s \frac{(s - 1)}{2} + s (2r + 1) $$ After failing at that, I tried with Wolfram Alpha, and got the answer I wanted. But, how did it get ...
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0answers
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Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions

Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i < j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to ...
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1answer
22 views

Generic points as coefficients of polynomial kernels?

I am reading the paper Dual-to-Kernel Learning with Ideals. Here is part of it: The definition/motivation of genericity in Wikipedia are A generic point of the topological space $X$ is a point ...
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2answers
44 views

Remainder of the polynomial

A polynomial function $f(x)$ with real coefficients leaves the remainder $15$ when divided by $x-3$, and the remainder $2x+1$ when divided by $(x-1)^2$. Then the remainder when $f(x)$ is divided by ...
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1answer
56 views

Working out the discriminant to a polynomial and using for working out “a”

For an equation: $$ x-b^2/x^3+a=0 \\$$ i.e. $$ x^4-b^2+ax^3=0 \\$$ If the discriminant is positive (i.e. $> or =0$) for real roots, what is the discriminant for these equations? Can you use the ...
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1answer
43 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
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0answers
20 views

Show equivalence corresponding Nulls of function.

I'd like to show that the following two propositions are equivalent: (1) $f \in \mathbb{R}[x]$ has a multiple Null, so it's $\ge 2$. (2) $f$ and $f'$ have a common Null, whereas $f'$ describes the ...
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2answers
15 views

Estimate the error of interpolating ,,,

Estimate the error of interpolating (${lnx}$) . at ${x=3}$ with an interpolation polynomial with base points ${x=1 , x=2 , x=4 , x=6 }$ .
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2answers
52 views

$f$ has root $\alpha$, then $f = (X-\alpha)g$ for some $g$

I need some help with the following problem: Suppose $R$ is a unique factorisation domain and $f \in R[X]$ such that $\deg f > 0$ and $f$ has a root $\alpha \in R$. Then $f = (X-\alpha)g$ for some ...
4
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2answers
91 views

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real?

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real? The polynomial, call it $f(x)$ must be irreducible over $\mathbb{Q}$ (or over $\mathbb{Z}$ as Gauss' lemma can be used.) ...
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3answers
93 views

How to factor the polynomial $x^4-x^2-2x-1$?

By taking advantage of a computer algebra system, I found out that $$x^4 - x^2 - 2x - 1 = (x^2 + x + 1)(x^2 - x - 1)$$ However, I don't know a straightforward way to solve this by hand. This was in ...
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3answers
49 views

How to find number of real and complex roots?

Below is a question asked in JNU Entrance exam for M.Tech/PhD. I want to know if there is a fixed way to calculate it. I have failed to use the factor theorem. ...
2
votes
3answers
103 views

Find the polynomial ${P(z)}$ of degree ${3}$ such that …

I meant that if we have ${P(z)}$ of degree ${3}$ such that ..... $${P(-1)=7} , {P(2)=3} ,{P(4)=-2} ,{P(6)=8}$$ Find the polynomial
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1answer
46 views

Solutions for $a$ by factoring a multivariate polynomial

I have an equation: $$\left(\frac{b}{x^2}+1\right)⋅\left(x−\frac bx\right)+a=0$$ The question is by factorizing what are the solutions for a? I am not sure how to do this: I have reduced the ...
0
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1answer
26 views

minimal polynomial and linear transformation

If $T:\Bbb{C} \to \Bbb{C}$ defined by $T(x)=x$ . T satisfity minimal poly is $x-1$. Is it correct. Any polynomial of degree $>1$ is a linear transformation on C .this type of transformation exist ...