Polynomial Definition and 1000 Threads

Homework Statement Solve for the roots of the following. (What do you notice about the complex roots?) b) x3 + x2 + 2x + 1 = 0 Homework Equations To find roots of a polynomial of degree n > 3, look at the constant and take all its factors. Those are possible roots. Then plug them into see...

Homework Statement Determine if the following is a subspace of ##P_3##. All polynomials ##a_0+a_1x+a_2x^2+a_3x^3## for which ##a_0+a_1+a_2+a_3=0## Homework Equations use closure of addition and scalar multiplication The Attempt at a Solution Let ##P=a_0+a_1x+a_2x^2+a_3x^3## and...

Homework Statement z2-(3+i)z+(2+i) = 0 Homework EquationsThe Attempt at a Solution [/B] Does the quadratic formula work in this case? Should you deal with the real and complex parts separately?

Hey! :o Let $f :\rightarrow \mathbb{R}$, $f(x) := tan(x)$. I want to find a $N\in \mathbb{N}$ such that for the $N$-th Taylor polynomial $P_N$ at $0$, that is defined as follows $P_N(x)=\sum_{n=0}^N\frac{f^{(n)}(0)}{n!}x^n$, it holds that $$\left |f(x)-P_N(x)\right |\leq 10^{-5}, \ \ x\in...

Homework Statement Given the polynomial function ##x^4+x^3+2x^2+4=0## solve it if you know that it has at least one complex zero whose real part equals the complex part. Homework Equations 3. The Attempt at a Solution [/B] My guess is that if this function has one complex zero it must have a...

Assume that (x-4) is a factor of 2x^2-4x-z. What is the value of z? How would you set it up to use the foil method?

Factor $30x^4-41x^3y-129x^2y^2+100xy^3+150y^4$. Please help me get started. I tried grouping the terms but still can't see any factorization that is familiar to me. Thanks.

I'm just curious as to how to go about factoring a polynomial like this one $6x^4+17x^3-24x^2-53x+30$ without using rational root theorem? Thanks

Please assist me in this problem.$\frac{2}{3}b^5-\frac{1}{6}b^3+\frac{4}{9}b^2-1$ I tried grouping but still could not find anything factorable form of the expression. Regards.

Homework Statement Prove that if p(x) has even degree with positive leading coefficient, and ## p(x) - p''(x) \geq 0 ## for all real x, then $$ p(x) \geq 0$$ for all real x Homework Equations N/A Problem is from Art and Craft of Problem Solving, as an exercise left to the reader following a...

Evaluate number of real roots of the equation $$x^6-x^5+x^4-x^3+x^2-x+\frac{2}{5} = 0$$

Hey! :o In my notes there is the following: Let $F$ be a field. The irresducible $f\in F[x]$ is separable, if all the roots are different. A non-constant polynomial $f\in K[x]$ is separable, if all the irreducible factors are separable. Example: $f(x)=(x^2-2)^2(x^2+3)\in \mathbb{Q}[x]$...

Homework Statement I would like to show that if ##p(x) = \sum_{i=1}^m a_i x^i## and ##q(x) = \sum_{j=1}^n b_j x^j##, then ##p(x)q(x) = \sum_{k=0}^{m+n} \left( \sum_{i+j=k} a_i b_j \right) x^k##, where the polynomial ring is assumed to be commutative. Homework EquationsThe Attempt at a Solution...

It is required to determine if there is a non-constant polynomial p with positive coefficients such that function $x \mapsto p(x^2)-p(x)$ is decreasing on $[1,+\infty \rangle$. What should I do here? How should I exactly determine that? What is the right method? My idea was to use somehow the...

Homework Statement Show that the polynomial f(x)=x^5-3x^4+6x^3+18x^2-3 is NOT solvable by radicals Homework EquationsThe Attempt at a Solution I'm pretty sure that to prove that this polynomial is not solvable I am too show that it has exactly 3 roots. That means that it will have 3 roots and...

Homework Statement Show that the polynomial f(x) = x^5 - x^3 - 3x^2 + 3 is solvable by radicals where the coefficients of f are from the field of rational numbers. Homework EquationsThe Attempt at a Solution My strategy to solve this problem was to construct a splitting field and then see if...

Homework Statement f(x) = x^7 + 3x^6 + 3x^5 - x^3 - 3x^2 - 3x where the coefficients are elements of F_5. Show that this polynomial is divisible by x^5-x and construct a splitting field L for f over F_5 and computer [L:F_5] Homework EquationsThe Attempt at a Solution So the first thing I did...

Homework Statement Find the splitting field of x^9-1 over F_13 (the field of 13 elements) Homework EquationsThe Attempt at a Solution Every element in the cyclic group F_13* will have order 13 since 13 is prime, and thus 1 is the only root of x^9-1 in F_13. Thus I did the long vision of...

Hello Everybody , First of all, I would like to apologize that this problem contains 3 parts to it (3 questions) but they all relate to each other. You must complete one part to move on to the next part. With that being said, I have 3-part problem that I could use some assistance with. 1a...

I'm currently studying the sensitivity of polynomial roots as a function of coefficient errors. Essentially, small coefficient errors of high order polynomials can lead to dramatic errors in root locations. Referring to the Wilkinson polynomial wikipedia page right...

Homework Statement Find the roots of x^4 - 6x^2 - 2 Homework EquationsThe Attempt at a Solution So my first observation is that this polynomial is irreducible by Eisenstein criterion with p=2. If I substitute y=x^2 then this polynomial becomes a quadratic, and I can apply the quadratic...

Hey! :o Let $E/F$ be a finite extension. I want to show that this extension is Galois if and only if $E$ is a splitting field of a separable polynomial of $F[x]$. I have done the folllowing: $\Rightarrow$ : We suppose that $E/F$ is Galois. So, we have that the extension is normal and...

$\tiny{206.11.1.12}$ $\textsf{a.Find the linear approximating polynomial for} \\$ $$\displaystyle f(x)=\cos{x} \textsf{ centered at $\displaystyle a=\frac{\pi}{4}$.} \text{approximate} \cos(0.28\pi)$$ $\textsf{ using}$ $$f(x)=f(a)+f'(a)(x-a)$$ $\textsf{b. Find the quadratic approximating...

Say that we have a sequence defined by the mth degree polynomial, ##a_n=\displaystyle \sum_{k=0}^{m}c_kn^k##. I found the following formula which is a recursive representation of the same sequence: ##\displaystyle a_n =\sum_{k=1}^{m+1}\binom{m+1}{k} (-1)^{k-1}a_{n-k}##. I'm curious as to why...

Homework Statement The task is to find the extreme values (and their nature) of the polynomial function . $$f(\vec{x})=x_1x_2+x_1^2+x_2^2+x_3^3+x_4^4.$$ The Attempt at a Solution The critical point is ##a=(0,0,0,0)##, which is the solution to ##\nabla{f(a)}=0.## If we form the Hessian matrix...

So, I've got an assignment to prove that f(x)=\cos{(n \cdot \arccos{x})} is a polynomial for \forall n \in \mathbb{N} . Also, we were suggested to use mathematical induction. So, I've tried: Base step: n=1 \implies f(x)=\cos{(\arccos{x})}=x Assumption step: f(x)=\cos{(n \cdot \arccos{x})}...

Consider the following integration: $$\int \frac{d^{4}k}{(2\pi)^{4}}\ \frac{1}{(k^{2}+m^{2})^{\alpha}}=\frac{1}{(4\pi)^{d/2}} \frac{\Gamma\left(\alpha-\frac{d}{2}\right)}{\Gamma(\alpha)}\frac{1}{(m^{2})^{\alpha-d/2}}.$$ --- How does the dependence on ##d## arise in this integral? Can someone...

Homework Statement y=1/(x^1/4). I'm given 5 x's and 5y's. I need to write Newton interpolating polynomial and find the error. Homework Equations Ln-1(x)=f(x1)+f(x1,x2)(x-x1)+... The Attempt at a Solution With the formula above I wrote the Newton interpolating polynomial but I can't find the...

Homework Statement Let a = (1+(3)^1/2)^1/2. Find the minimal polynomial of a over Q. Homework EquationsThe Attempt at a Solution Maybe the first thing to realize is that Q(a):Q is probably going to be 4, in order to get rid of both of the square roots in the expression. I also suspect that...

Homework Statement So I need the find the minimal polynomial of the primitive 15th root of unity. Let's call this minimal polynomial m(x) Homework EquationsThe Attempt at a Solution I know that m(x) is an irreducible factor of x^15 - 1 and also that the degree of m(x) is equal to the Euler...

Hey! :o Let $F$ be a field, $D=F[t]$, the polynomial ring of $t$, with coefficients from $F$ and $K=F(t)$ the field of rational functions of $t$. (a) Show that $t\in D$ is a prime element of $D$. (b) Show that the polynomial $x^n-t\in K[x]$ is irreducible. (c) Let $\text{char} F=p$. Show...

Homework Statement Show that x^7 + 3x^6 + 3x^5 - x^3 - 3x^2 - 3x is divisible by x^5-x Homework EquationsThe Attempt at a Solution So i did polynomial long division and as a quotient so far I have x^2+3x, and it appears that my remainder is going to be 3(x^3-x). Does this mean that I did...

Hi, I am struggling with the following problem: "Let $V=P_3(\Bbb{R})$ and let $t_1=3x^3-x-2$ and $t_2=x^3-3x+2$ with $T=\left\{ t\in V \:|\: t(1)=0 \right\}$. Find ${t_3}\in\left\{T\right\}$ such that $\left\{t_1, t_2, t_2\right\}$ is a basis of T. Not sure where to go as each column matrix...

Homework Statement Find the minimal polynomial of a = i*(2)^1/2 + (3)^1/2 Homework EquationsThe Attempt at a Solution Well, I know the minimal polynomial will have degree four, and that's about it. Will it help if I look at the linear factors of the minimal polynomial in some splitting field...

Homework Statement Construct a splitting s for the polynomial x^3+2x+1 over Z/Z3 Homework Equations 4=1 Mod 3 :P The Attempt at a Solution So I'm actually quite confused. There are no roots for x+3+2x+1 over Z/Z3. I am used to constructing splitting fields with polynomials that have...

I have the following question: Is there a basis for the vector space of polynomials of degree 2 or less consisting of three polynomial vectors ##\{ p_1, p_2, p_3 \}##, where none is a polynomial of degree 1? We know that the standard basis for the vector space is ##\{1, t, t^2\}##. However...

Hey! :o Let $f = x^4−2x^2−1 \in \mathbb{Q}[x]$. We have that $f(x+1)=(x+1)^4-2(x+1)^2-1=x^4+4x^3+6x^2+4x+1-2(x^2+2x+1)-1=x^4+4x^3+4x^2-2$ We have that $p=2$ divides all the coefficients $4,4,-2$ and $p^2=4$ does not divide the constant term $-2$. So, the polynomial $f(x+1)$ is Eisenstein...

The polynomial equation and it's private solution: $$(1)~~ay''+by'+cy=f(x)=kx^n,~~y=A_0x^n+A_1x^{n-1}+...+A$$ If i, for example, take ##f(x)=kx^3## i get, after substituting into (1), an expression like ##Ax^3+Bx^2+Cx+D## , but that doesn't equal ##kx^3##

Homework Statement Help i have a homework quiz done and i simply can't find out how to do the 3rd problem as we haven't even learned how to do it or maybe my notes aren't good or something , however I am close to an A in the class and this would help bring it closer. It asks me: "Find all the...

I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or...

Homework Statement Solve the following. Express answers in set notation. -2(x-2)(x-4)(x+3)<0 Homework EquationsThe Attempt at a Solution I know my four intervals are x<-3 , -3<x<2 , 2<x<4 , x>4. I thought the answer would be x<-3 and 2<x<4 however the answers are opposite of what I thought...

The polynomial: $P(x) = 1 + a_1x +a_2x^2+...+a_{n-1}x^{n-1}+x^n$ with non-negative integer coefficients has $n$ real roots. Prove, that $P(2) \ge 3^{n}$

I am stuck with one proof and I need some help because I don't have any idea how to proceed at this moment. The task says: If f(x) is a polynomial with integer coefficients, and if f(a)=f(b)=f(c)=-1, where a,b,c are three unequal integers, the equation f(x)=0 does not have integer solutions...

Homework Statement Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms. Degree 4: No real zeros, complex zeros of 1+i and 2-3i Homework Equations (-b±√b^2-4ac)/2a The Attempt at a Solution I want...

Homework Statement The collar A slides on the vertical smooth bar. Masses ma=20 kg, mb = 10 kg, and spring constant k = 250 kN/m. When h = 0.2m, the spring is unstretched. Determine the value of h when the system is at rest. Homework Equations sum of all forces equal zero sum of all moments...

Homework Statement A polynomial, P(x), is fourth degree and has all odd-integer coefficients. What is the maximum possible number of rational solutions to P(x)=0? Homework Equations P(x) = k(x-r1)(x-r2)(x-r3)(x-r4) P(x) = 0 when x = {r1, r2, r3, r4} The Attempt at a Solution I expanded the...

Let P(x) be a polynomial of x. Show that there exists a polynomial Q(x) such that P(x)Q(x) is a polynomial of $x^3$

First Question: Solve the following system of equations log{x+1}y=2 log{y+1}x=1/4 Work: Turned them into equations (x+1)^2=y (y+1)^(1/4)=x Substituted second equation into the first equation ((y+1)^(1/4)+1)^2=y factored out and eventually got ((y+1)^1/4)^2+2((y+1)^1/4)+1=y Tried...

Hi The polynomial ( 1+x^2 ) Can this polynomial span the space of polynomials of degree 2 in standard basis ?

I am examining the polynomial approximation for $e^x$ near $x = 2$. From Taylor's theorem: $$e^x = \sum_{n = 0}^{\infty} \frac{e^2}{n!} (x - 2)^n + \frac{e^z}{(N + 1)! } (x - 2)^{N - 1}$$ Now, I don't get the next part: We need to keep $\left| (x - 2)^{N + 1} \right|$ in check so we can...