Polynomial Definition and 1000 Threads

Homework Statement Given the matrix 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 What is the minimal polynomial? Homework Equations - The Attempt at a Solution This is the Jordan form, so I guess the solution is just...

In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The new challenge: Are there any...

In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The challenge: Prove that the...

Homework Statement Determine the least possible degree of the function corresponding to the graph shown below. Justify your answer. Homework Equations The graph is attached. I remade the graph using google grapher, but the graph I got in the test have exactly the same x-intercepts (-2 of order...

so I was reading my textbook and was showing steps on applying the quotient rule to the function: y=ex/(1+x2) it went from (1+x2)(ex)-(ex)(2x)/(1+x2)2 to ex(1-x)2/(1+x2)2 I understand the first step, but don't get how they got to ex(1-x)2 in the numerator. can someone please explain the...

Not sure how this can be done. can anyone help?

Homework Statement I have found the roots of my polynomial: ## (2x+3y)^{2}-1 =0 ## Roots are x=3n+2 & y=-2n-1, where n belongs to all Z. What does it mean that the solution has arbitrary large coordinates? The Attempt at a Solution I think I know the basic concept of root. It could be...

Homework Statement 1. Let ##p(x) = a_{0} x^{n} + a_{1} x^{n−1} + ... + a_{n} , a_{0} \neq 0 ##be an univariate polynomial of degree n. Let r be its root, i.e. p(r) = 0. Prove that ## |r| \leq max(1, \Sigma_{1 \leq i \leq n} | \dfrac{a_{i} }{ a_{0} } | )## Is it always true that? ## |r| \leq...

Homework Statement Find a specific number δ>0 such that if x2 + y2 = δ2, then |x2+y2+3xy+180xy5 < 1/10 000. Answer: Choose δ < 0.002 Homework Equations ε-δ def'n of limit: lim (x,y) → (a, b) f(x) = L if for every ε > 0 there exists a δ > 0 such that 0 < √(x-a)2+(y-b)2, |f(x) - L| < ε...

In Dummit and Foote Chapter 13: Field Theory, the authors give several examples of field extensions on page 515 - see attached. In example (3) we read (see attached) " (3) Take F = \mathbb{Q} and p(x) = x^2 - 2 , irreducible over \mathbb{Q} by Eisenstein's Criterion, for example" Now...

Homework Statement A matrix A\inMn(ℂ) is diagonalizable if and only if mA(x) has no repeated roots. Homework Equations If A\inHom(V,V) = {A:V→V | A is a linear map}, the minimal polynomial of A, mA(x), is the smallest degree monic polynomial f(x) such that f(A)=0. The Attempt at a...

Hopefully, anyone who has studied polynomial long division understands the link between it and regular long division. If you divide $58$ into $302985$, you could follow the usual long division procedure and obtain the answer. Alternatively, if you divide $x+4$ into $x^{3}+2x^{2}+6x+7$, you could...

Hello Everyone, Actually i would like to have a clear Explanation about Runge Phenomenon. Could you help me Please.

Homework Statement You have a fixed payment loan, you know the quantity you need to pay every year, the years until maturity and (I suppose to) the loan value and you need to calculate which is the yield to maturity. Homework Equations Loan Value =Fp/(1+i) + Fp/(1+i)2+...+ Fp/(1+i)n...

A polynomial P(x) when divided by(x-1) leaves a remainder 1 and when divided by (x-2) leaves a remainder of3. prove that when divided by(x-1(x-2) it leaves a remainder -2x=5. thank you.

Problem: q(x)=x^2-14\sqrt{2}x+87. Find 4th degree polynomial p(x) with integer coefficients whose roots include the roots of q(x). What are the other two roots of p(x)? I found that the two roots of q(x) are x=7\sqrt{2}-\sqrt{11} and x=7\sqrt{2}+\sqrt{11}. Since they are conjugates of...

Hi! Does anybody know if there is something in common between Bernstein functions and Bernstein polynomials except the word 'Bernstein'? I mean from mathematical point of view.

1. Hi I have a question I am stuck on it is: (x3 + 2x2y - 2xy2 - y3)/(x-y) Can anyone help? Homework Equations The Attempt at a Solution

Let $p(x)=x^m-1$ be a polynomial over $\mathbb Q$ and $E$ be the splitting field for $p$ over $\mathbb Q$. We know that $p$ has $\phi(m)$ primitive roots in $E$, where $\phi$ is the Euler's totient function. Let $\omega$ be a primitive root of $p$. Define $\theta_k:E\to E$ as...

Prove that there are only two real numbers such that $(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=720$.

Find the real root of x^5+5x^3+5x-1

How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?

Similar matrices share certain properties, such as the determinant, trace, eigenvalues, and characteristic polynomial. In fact, all of these properties can be determined from the character polynomial alone. However, similar matrices also share the same rank. I was wondering if the rank is...

Let $L$ be an extension of a field $F$. Let $\alpha_1, \alpha_2\in L$ be such that both of them are algebraic over $F$ and have the same minimal polynomial $m$ over $F$. We know that there is an isomorphism $\phi:F(\alpha_1)\to F(\alpha_2)$ defined as $\phi(\alpha_1)=\alpha_2$ and $\phi(x)=x$...

Homework Statement Factorize : (x+1) (x+2) (x+3) (x+6)-3 x2 Homework Equations - The Attempt at a Solution Expanding everything , I get x4+12x3+44x2+72x+36 . At this point I tried few guesses using rational roots test. But it appears this has no rational roots. So how should...

The product of two of the roots of the equation ax^4 + bx^3 + cx^2 + dx + e = 0 is equal to the product of the other two roots. Prove that a*d^2 = b^2 * e

Obtain the sum of the squares of the roots of the equation x^4 + 3x^3 + 5x^2 + 12x + 4 = 0 . Deduce that this equation does not have more than 2 real roots . Show that , in fact , the equation has exactly 2 real roots in the interval -3 < x < 0 . Denoting these roots α and β , and the other...

The roots of the equation x^3 - x - 1 = 0 are α β γ and S(n) = α^n + β^n + γ^n (i) Use the relation y = x^2 to show that α^2, β^2 ,γ^2 are the roots of the equation y^3 - 2y^2 + y - 1 =0 (ii) Hence, or otherwise , find the value of S(4) . (iii) Find the values of S(8) , S(12) and S(16)I have...

Find the sum of the squares of the roots of the equation x^3 + x + 12 = 0 and deduce that only one of the roots is real . The real root of the equation is denoted by alpha . Prove that -3< alpha < -2 , and hence prove that the modulus of each of the other roots lies between 2 and root 6 . I...

Homework Statement Factorise: f(x)=x^3-10x^2+17x+28

Homework Statement Show that if ## a > 0##, ##ax^2 + 2bx +c >= 0 ## for all values of ##x## iff ##b^2 -ac <=0 ## Homework Equations The Attempt at a Solution Well, I don't think this really makes any sense but away we go. All I did was take ##ax^2 + 2bx +c >= 0 ## and...

$a,b,c \in N$ $c+1=2a^2$ $c^2+1=2b^2$ c is a prime a,b,c are roots of $ 3x^3+rx^2+sx+t=0 $ please find r and t

Hi there, I was wondering if it is possible to find the roots of the following polynomial P(x)=x^n+a x^m+b or at least can I get the discriminant of it, which is the determinant of the Sylvester matrix associated to P(x) and P'(x). Thanks

I've been looking for proof of the fact that the characteristic polynomial of an n by n matrix has degree n with leading coefficient ## (-1)^{n} ##. I first tried proving it myself but my method is a bit strange (it does use induction though) and I am doubting the rigor, so could perhaps...

Homework Statement Write each polynomial as the product of it's greatest common monomial factor and a polynomial Homework Equations 8x^2+12x 6a^4-3a^3+9a^2 The Attempt at a Solution 4x(2x+3) a^2(6a^2-3a+9) I really don't understand what it's asking me for, how can I factor this...

The problem statement Let V be a vector space of dimension 8 and f (endomorphism) such that the minimal polynomial of f is x^​7. If B={v1,...,v8} is the Jordan basis of f, find the Jordan form and a Jordan basis for f^​2 and f^​3. The attempt at a solution Ok, I am having some trouble to...

Find the constants a,\;b, \;c,\; d such that 4x^3-3x+\frac{\sqrt{3}}{2}=a(x-b)(x-c)(x-d).

3.) ∫(3+s)1/2(s+1)2ds

1. ∫(x2-4x+4)4/3 2. ∫(1+1/3x)1/2dx/x2 this is what i do for number 1. ∫(x2-2)8/3 now I'm stuck. please help!

Find the polynomial of the lowest degree with integer coefficients such that one of its roots is \sqrt{2} + \sqrt[3]{3}.

When we write F[x_1, x_2, ... ... , x_n] where F is, say, a field, do we necessarily mean the set of all possible polynomials in x_1, x_2, ... ... x_n with coefficients in F? [In this case, essentially all that is required to determine whether a polynomial belongs to F[x_1, x_2, ... ... ...

Hi all, I would like to find the distribution (CDF or PDF) of a random variable Y, which is written as Y=X_1*X_2*...X_N/(X_1+X_2+...X_N)^N. X_1, X_2,...X_N are N i.i.d. random variables and we know they have the same PDF f_X(x). I know this can be solved by change of variables technique and...

x^3-5x-6=0 i've tried the p/q calculations in accordance with the rational roots theorem but I've yet to find the answers...

Two polynomial f(x) and g(x) are equal then their degrees are equal. This is a very trivial statement and it shouldn't worry me much but it is. I get an intuitive idea why they should be equal. Their graphs wouldn't coincide for unequal degrees. But what if somehow the coefficients make...

Homework Statement p(x) = x^4+10x^3+26x^2+10x+1 p(x) = a(x)b(x) where a(x) and b(x) are quadratic polynomials with integer coefficients. It is given that b(1) > a(1). Find a(3) + b(2). Homework Equations p(x) = x^4+10x^3+26x^2+10x+1The Attempt at a Solution I tried to factor the given quartic...

If a,\;b are roots of polynomial x^4+x^3-1, prove that a(b) is a root of polynomial x^6+x^4+x^3-x^2-1.

Let A be a square matrix, a) show that (I-A)^{-1}= I + A + A^2 + A^3 if A^4 = 0 b) show that (I-A)^{-1}= I + A + A^2+...+A^n if A^{n+1}= 0

Solve x^4+1=2x(x^2+1).

Homework Statement Find a polynomial f(x) of degree 4 which increases in the intervals (-∞,1) and (2,3) and decreases in the interval (1,2) and (3,∞) and satisifes the condition f(0)=1 Homework Equations The Attempt at a Solution Let f(x)=ax^4+bx^3+cx^2+dx+1 f'(1)=f'(2)=f'(3)=0 But...

I am reading the Proof of Hilbert's Basis Theorem in Rotman's Advanced Modern Algebra ( See attachment for details of the proof in Rotman). Hilbert's Basis Theorem is stated as follows: (see attachment) Theorem 6.42 (Hilbert's Basis Theorem) If R is a commutative noetherian ring, the R[x] is...