Polynomial Definition and 1000 Threads

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...

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...

Homework Statement Let x\inR, Prove that if x>2 then x^4 - 8x^3+24x^2-32x+16 Homework EquationsThe Attempt at a Solution So far I have only learned proofs involving even and odd numbers, that sort of thing. I'm not really sure how to approach this one. I was thinking that a proof by cases...

I am reading Dummit and Foote Section 9.5 Polynomial Rings Over Fields II and need some help and guidance with the proof of Proposition 15. Proposition 15 reads as follow: Proposition 15. The maximal ideals in F[x] are the ideals (f(x)) generated by irreducible polynomials. In particular...

Let $r(x)\in\mathbb Q(x)$ be a rational function over $\mathbb Q$. Assume $r(n)$ is an integer for infinitely many integers $n$. Then show that $r(x)$ is a polynomial in $\mathbb Q[x]$.

Homework Statement Let y(x) = a0 + a1x + a2x2 + a3x3 + a4x4 Represent the differential equation (d/dx)[ (1-x2)(dy/dx) ] + λy = 0 in matrix form. Find the values of λ for which there is a solution to the matrix equation, and find the solutions for the smallest and largest values of...

Homework Statement Prove the following For each real a, the function p given by p(x) = f(x+a) is a polynomial of degree n. Homework Equations Can I have a hint I have hard time starting.Thanks The Attempt at a Solution

I am reading Dummit and Foote Section 9.4 Irreducibility Criteria. In particular I am struggling to follow the proof of Eisenstein's Criteria (pages 309-310 - see attached). Eisenstein's Criterion is stated in Dummit and Foote as follows: (see attachment) Proposition 13 (Eisenstein's...

ny = x^(n-1) + x^(n-2) + ... + x + 1 for a certain y and n (>10000) with y!=1 and ny > 1. Is there an analytic way to solve this? Thank you.

I have asked a similar question but it wasn't answered fully so here it goes. Why is it that you only need two points to find a linear equation, three for a quadratic and so on. What determines this and is this determined because of the characteristics of these different degree polynomials?

Hey guys, I know what polynomials are but what I really don't understand is the way you are able to find the equation to a set amount of points. I don't understand why you have to have a certain amount of points to find different degrees of functions. For example, why are only three points...

Homework Statement Show by polynomial division that \frac{x^3-3x^2+12x-5}{x-2}=(x^2-x+10)+\frac{15}{x-2} The Attempt at a Solution Please see attachment

Homework Statement Use polynomial long division to determine the quotient when 3x^3-5x^2+10x+4 divided by 3x+1 The Attempt at a Solution Please see attachment as I wasn't quite sure how to write my answer here :shy:

I am reading Dummit and Foote Section 9.3 Polynomial Rings That are Unique Factorization Domains (see attachment Section 9.3 pages 303 -304) I am working through (beginning, anyway) the proof of Theorem 7 which states the following: "R is a Unique Factorization Domain if and only if R[x] is a...

Hi all. I need some advice in a project I'm into. I have some experimental (simulation) data and i need to find a function that fits to it. The experimental data behaviour change when I modify some parameters I have. My goal is, from that single function, been able to predict how the...

I've looked examples up online and I just can't figure out what to do exactly when I have (2x^2)/(x^2+1), for some stupid reason that was probably the work of satan, EVERY problem on the internet only has the lead coefficient of the numerator equal to or less than that in the denominator and...

Prove that if f(x) and g(x) are polynomials with rational co-efficients whose product f(x)g(x) has integer co-efficients, then the product of any co-efficient of g(x) with any coefficient of f(x) is an integer. My initial thoughts on this are that the exercise seems to be set up for an...

Exercise 1, Section 9.3 in Dummit and Foote, Abstract Algebra, reads as follows: Let R be an integral domain with quotient field F and let $$ p(x) \in R[x] $$ be monic. Suppose p(x) factors non-trivially as a product of monic polynomials in F[x], say $$ p(x) = a(x)b(x) $$, and that $$ a(x)...