Polynomial Definition and 1000 Threads

Homework Statement I did some refresher exercises in khan academy. I tried to wrap my head around the factoring problems, but it did not go well... I did not understand the khan academy tips and hints. (about factoring) one particular difficult polynomial was thus: factorize the polynomial...

I'm a programmer looking for a way to create polynomial equations from a list of x intercepts and local maxima. For the sake of discussion we can begin with a function of degree 4. The scale and position of the curve is unimportant so for simplicity's sake the curve can always have x intercepts...

Hey! :o We suppose that $M/L/K$ are consecutive fields extensions and $a\in M$ is algebraic over $K$. I want to show that $a$ is algebraic also over $L$. I want to show also that the minimal polynomial of $a$ over $L$ divides the minimal polynomial of $a$ over $K$ (if we consider this...

3(2k+1)3 I have written a program which calculates the value of that polynomial with different values of k. The result is always an odd number. I am having a difficult time writing a proof that states that this polynomial always returns an odd number. I know that (2k + 1) is the general form...

I derived a relation between the product of two linear polynomials and the square of their average. It can be used to solve any quadratic equation. Will this help me getting into a top university?

Suppose [K:F]=n, where K is a root field over F. Prove K is a root field over F of every irreducible polynomial of degree n in F[x] having a root in K. I don't believe my solution to this problem because I 'prove' the stronger statement: "K is a root field over F for every irreducible...

NOTE: presume real coefficients If a pair of polynomials have the Greatest Common Factor (GCF) as 1, it would seem that any root of one of the pair cannot possibly be a root of the other, and vice-versa, since as per the Fundamental Theorem of Algebra, any polynomial can be decomposed into a...

i am trying to divide x^3+(1-k^2)x+k by (x+k) but i can't do this can you show me how to.

Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers. Homework Statement I want to solve: $$ x+a(x^2-b)^{1/2}+c=0$$ Homework Equations The Attempt at a Solution My approach is to make a change of variable x=f(y) to get a true polynomial (integer...

Hi folks, I know the work from Galois showing it is not possible to solve certain equations using only a certain type of numbers. But that was more than 100 years ago, I suppose lots of progress has been made on this topic. So my question is, What has been discovered to solve polynomial...

Homework Statement A monic polynomial is a polynomial which has leading coefficient 1. Find the real, monic polynomial of the lowest possible degree which has zeros −1−2i,−2i and i. Use z as your variable.The Attempt at a Solution [/B] Would I just expand the zeros giving me...

Homework Statement [/B] Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero. The Attempt at a Solution [/B] (z-(1+i)(z-i) = Z^2-z-1-2iz+i (Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i) Z^2 term...

The attachment below proves that an nth-degree polynomial has exactly ##n## roots. The outline of the proof is as follows: Suppose (1.1) has ##r## roots. Then it can be written in the form of (1.8) by factor theorem. Next use the second fundamental result in algebra (SFRA): if ##f(x)=F(x)## for...

Given a polynomial ##f(x)##. Suppose there exists a value ##c## such that ##f(c)=f'(c)=0##, where ##f'## denotes the derivative of ##f##. Then ##f(x)=(x-c)^mh(x)##, where ##m## is an integer greater than 1 and ##h(x)## is a polynomial. Is it true? Could you prove it? Note: The converse is true...

Homework Statement p(x) = x^3 − x^2 + ax + b is a real polynomial with 1 + i as a zero, find a and b and find all of the real zeros of p(x).The Attempt at a Solution [/B] 1-i is also a zero as it is the conjugate of 1+i so (x-(1+i))(x-(1-i))=x^2-2x+2 let X^3-x^2+ax+b=x^2-2x+2(ax+d)...

I was reading about Gauss's Lemma here: https://cims.nyu.edu/~kiryl/Algebra/Section_3.10--Polynomials_Over_The_Rational_Field.pdf Unfortunately, I am stuck on Lemma 3.10.1 that concludes that the product of a pair of primitive polynomials is itself primitive. I understand about how there is...

Homework Statement Prove the following statement: Let f be a polynomial, which can be written in the form fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0 and also in the form fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0 Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n Homework Equations 3. The...

Homework Statement [/B] Polynomial f(x) is divisible by ##x^2-1##. If f(x) is divided by ##x^3-x##, then the remainder is... A. ##(x^2-x)f(-1)## B. ##(x-x^2)f(-1)## C. ##(x^2-1)f(0)## D. ##(1-x^2)f(0)## E. ##(x^2+x)f(1)## Homework Equations Remainder theorem The Attempt at a Solution [/B]...

Hi there, I'm working on a problem right now that relates to least squares error estimate for polynomial fitting. I'm aware of techniques (iterative formulas) for finding the coefficients of a polynomial that minimizes the square error from a data set. So for example, for a data set that I...

Homework Statement I have a series of questions regarding a cyclotomic polynomial of order p-1 where p is a prime; so there are p total terms in this polynomial because their is a constant term. I will post the questions 1 at a time, and as soon as I work my way through one i'll post the next...

Homework Statement (Z^3)+3i(conjugate z) = 0 find all solutions. Homework EquationsThe Attempt at a Solution How can i isolate Z Tried factoring out z, didn't came out good: z(z^2+3i*(Conjugate z)/z) ==> Right part equal to zero. couldn't factor anymore.

Hello, I am looking for approximated or exact solution of \begin{align} I = \int_R \exp(cx^3-ax^2+bx)dx \end{align} where $a,b,c$ are complex numbers defined as: \begin{align} c &= \frac{1}{3}i\pi\phi'''(t) \notag\\ a &= \dfrac{1}{2\sigma^2}-i\pi \phi''(t) = re^{i\varphi}~~\text{with}~~~ r =...

Homework Statement Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n. Homework EquationsThe Attempt at a Solution I tried different approaches such as...

Homework Statement Solve the equation for r,r>0,r<R. \frac{-2\pi R^3}{3}-\frac{8\pi r^2\sqrt{R^2-r^2}}{3}+2r^2R+\frac{2\pi}{3}R^2\sqrt{R^2-r^2}=\frac{2\pi R^3}{3} 2. The attempt at a solution After factoring, -2R(2\pi R^2-3r^2)=2\pi\sqrt{R^2-r^2}(4r^2-R^2). After squaring, 64\pi^2...

I want to keep this question conceptual and qualitative (for now). 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 the variable I'd like to solve for and P, a are just real constants. I was wondering whether or not I could use...

Hey so I'm new to my TI nspire cx, still getting the hang of it. I've been trying to figure out how to get my eigenvector values to be fractions instead of decimals when I calculate them on here. Also, when I find the polynomial roots I get back decimals instead of fractions. I would like to...

I'm struggling about finding a way to find the upper bound of the error of Taylor polynomial approximation. I will explain better using a solved example I found... > $f: ]-3;+\infty[ \rightarrow \mathbb{R} $ $f(x)=ln(x+3) +1 $ >Find the upper bound of the error approximating the function in...

In the second paragraph after the expression of ##f(u)## below, it wrote "there are three roots to a cubic equation and three combinations of solutions". However, the combination of having three equal real roots was not mentioned. Why? In the next paragraph, in the second sentence, it wrote "at...

Question This is what I have done so far. I was wondering if anyone could verify that I found the correct minimum polynomial and roots? If I am incorrect, could someone please help me by explaining how I would find the min polynomial and roots? Thank you.

Homework Statement [/B] Th value of 'a' for which the equation x3+ax+1=0 and x4+ax+1=0 have a common root is?Homework EquationsThe Attempt at a Solution i initially thought of subtracting both the equations and then finding x and substituting back in the equation but it did not work.

Homework Statement Excel data for an assignment I'm doing has spit out a curve from some experimental data as shown here: http://i.imgur.com/KcJyEEj.png I'm wondering if there's a nice way to put this as a transfer function in the form of Y/X or something similar Homework EquationsThe...

In chapter 7 of Spivak calculus, it is proved that if n is odd, then the 'n'th degree polynomial equation f(x) has a root. I do understand what goes into the proof and can follow steps easily. But, my question is 1.How did they think of a proof like that? 2.By trial and error, did they find...

The problem asks for the diagonalization of (a(p^2)+b(x^2))^n, where x and p are position and momentum operators with the commutation relation [x,p]=ihbar. a and b are real on-zero numbers and n is a positive non-zero integer.I know that it is not a good way to use the matrix diagonalization...

Homework Statement Hello guys, I have a set of data containing x and y coordinates(width and length) as well as a 'z' coordinate that represents power density at each point of x and y given. I was hoping that someone might be able to help me figure out a way that I can find a function for z in...

Homework Statement Using the Generating function for Legendre polynomials, show that: ##P_n(0)=\begin{cases}0 & n \ is \ odd\\\frac{(-1)^n (2n)!}{2^{2n} (n!)^2} & n \ is \ even\end{cases}## Homework Equations Generating function: ##(1-2xt+t^2)^{-1/2}=\displaystyle\sum\limits_{n=0}^\infty...

Find the number of distinct real solutions of the equation $(x − 1)(x − 3)(x − 5) · · · (x − 2017) = (x − 2)(x − 4)(x − 6) · · · (x − 2016)$.

Hey all, I seek to find where the derivative of a nth order polynomial is at a 0, so far I have used secant method to find it, which works, but issue is is that that returns only one root, sliding the interval could work, but then itd always point to the edge of the interval, any help...

Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...

Let $1\lt a \lt 2$, $a$ is a root of the equation $x^5-x-2=0$. Prove that $\large a>\sqrt[9]{8}$.

Homework Statement Suppose a field F has n elements and F=(a_1,a_2,...,a_n). Show that the polynomial w(x)=(x-a_1)(x-a_2)...(x-a_n)+1_F has no roots in F, where 1_f denotes the multiplicative identity in F. Homework EquationsThe Attempt at a Solution Strategy: We have this polynomial...

Homework Statement Prove that for all n ≥ 1, there exists a polynomial f(x) ∈ ℚ[x] with deg(f(x)) = n such that f(x) is irreducible in ℚ[x]. Homework Equations In mathematics, a rational number is any number that can be expressed as the quotient or fractionp/q of two integers, a numeratorp...

Hello, Given the quartic: $$16x^4-40ax^3+(15a^2+24b)x^2-18abx+3b^2 = 0$$ where $a,b$ are certain real constants. My question is if there is a (simple) condition on $a$ and/or $b$ such that the quartic has at least one real root. Since the quartic has real coefficients the only possibilities...

This is a question that came about while I attempting to prove that a simple extension was a splitting field via mutual containment. This isn't actually the problem, however, it seems like the argument I'm using shouldn't be exclusive to my problem. Here is my attempt at convincing myself that...

Hello What is the standard definition of a polynomial? according to the book I am using a polynomial is an algebraic expression which is integral and rational for all the terms. It gives no definition of integral or rational seperately, but I think integral means that the variables are to...

Dear PF Forum, As we know in polynomial 2 degrees AX2 + BX + C = 0, there's a formula for solving it. What about 3 degrees for example: AX3 + BX2 + CX + D = 0, there's is really no formula for solving it? The only way to solve it is by hand? I have several methods in my head, at least...

Hi all, I'm hoping someone can help me out with this problem. I am following an example of a previous report to analyse some test results, but am having trouble with this step. The tests plot moment-rotation curves of a horizontal member fixed into a vertical member. You run 5 tests, and for...

Hi, I don't understand how to get to the answer of the question. The cubic equation x^3 + 3(x^2) +2 =O.by using substitution X=1/(u^0.5) get 4u^3+12u^2+9u-1 =0. I can't see where the 12 comes in It's question 10i

Let F=Z2 and let f(x) = X^3 +x+1 belong to F[x]. Suppose that a is a zero of f(x) in some extension of F. Using the field created above F(a) Show that a^2 and a^2+a are zeros of x^3+x+1?

Homework Statement Solve the ODE: y''+x*y'-y=0 Homework EquationsThe Attempt at a Solution Since this is a variable coefficient ODE, I have used the method of reduction of order, and assumed the solution in the form: y=c1*y1+c2*y2 In this case: y1=x, and I have the reached the integral below...

How is it implied?