Polynomial Definition and 1000 Threads

Let $a$ and $b$ be real numbers such that $a\ne 0$. Prove that not all the roots of $ax^4+bx^3+x^2+x+1=0$ can be real.

For ##x=-1## to be an *horizontal* inflection point, the first derivative ##y'## in ##-1## must be zero; and this gives the first condition: ##a=\frac{2}{3}b##. Now, I believe I should "use" the second derivative to obtain the second condition to solve the two-variables-system, but how? Since...

I was looking at this discussion of swapping roots of a polynomial causing the discriminant to loop around the origin. https://www.akalin.com/quintic-unsolvability Although it appears to be the case, has this mathematical fact ever been proven? It seems that the formula for the discriminant...

I'll write my considerations which lead me to get stuck on the ##\infty-\infty## form. $$\lim_{x \to +\infty }\sqrt{x^{2}-2x}-x+1 \rightarrow |x|\sqrt{1-0}-x+1$$ And I have no idea on how to go on...

I got this function in a function analysis and got confused on how to solve its positivity; I rewrote it as: $$\sqrt{x^{2}-2x}>x-1 \rightarrow x^2-2x>x^2-2x+1$$ And therefore concluded it must've been impossible... but I'm certainly missing something stupid, since plotting the graphs of the two...

if x_{I}, I = {1,2,...,2019} is a root of P(x) = ##x^{2019} +2019x - 1## Find the value of ##\sum_{1}^{2019}\frac{1}{1-\frac{1}{X_{I}}}## I am really confused: This polynomial jut have one root, and this root is x such that 0 < x < 1, so that each terms in the polynomial is negative. But the...

I often encounter functions called "polynomial" in numerous fields. I don't see an obvious common trait other than that they're usually describing a real-valued continuous function. What aspects are typical or universal or distinct? What structures can be polynomial? Some sources say that...

I am trying to prove the following expression below: $$ \int _{0}^{1}p_{l}(x)dx=\frac{p_{l-1}(0)}{l+1} \quad \text{for }l \geq 1 $$ The first thing I did was use the following relation: $$lp_l(x)+p'_{l-1}-xp_l(x)=0$$ Substituting in integral I get: $$\frac{1}{l}\left[ \int_0^1 xp'_l(x)dx...

Am I(always) legitimized to write ##-(a-b)^n=(b-a)^n##? I don't know why but it's confusing me... can't really understand when and why I can use that identity

Let $a,\,b,\,c$ be three distinct integers and $P$ be a polynomial with integer coefficients. Show that in this case the conditions $P(a)=b,\,P(b)=c,\,P(c)=a$ cannot be satisfied simultaneously.

I have problem solving this. The question is #19 in the first attachment. My work is in the second attachment I work to this point and get stuck: https://www.physicsforums.com/attachments/270843 Can anyone help me? Thanks

Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for every real $x$, then $p(x)\ge 0$ for every real $x$.

##x^{2017} + 1 = Q(x) . (x-1)^2 + ax + b## where ##Q(x)## is the quotient and ##ax+b## is the remainder ##x=1 \rightarrow 2 =a+b## Then how to proceed? Thanks

I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...

Note: $P_n (x)$ is legendre polynomial $$P_{n+1}(x) = (2n+1)P_n(x) + P'_{n-1}(x) $$ $$\implies P_{n+1}(x) = (2n+1)P_n(x) + \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} (2(n-1-2k)+1)P_{n-1-2k}(x))$$ How can I continue to use induction to prove this? Help appreciated.

Prove that there are no integers $a,\,b,\,c$ and $d$ such that the polynomial $ax^3+bx^2+cx+d$ equals 1 at $x=19$ and 2 at $x=62$.

Let $P$ be a real polynomial of degree five. Assume that the graph of $P$ has three inflection points lying on a straight line. Calculate the ratios of the areas of the bounded regions between this line and the graph of the polynomial $P$.

##f(x)## is divisible by ##(x-1) \rightarrow f(1) = 0## ##f(x) = Q(x).(x-1)(x+1) + R(x)## where ##Q(x)## is the quotient and ##R(x)## is the remainderSeeing all the options have ##f(-1)##, I tried to find ##f(-1)##: ##f(-1) = R(-1)## I do not know how to continue Thanks

The polynomial $x^3-kx+25$ has three real roots. Two of these root sum to 5. What is $|k|$?

If the equation $ax^2+(c-b)x+e-d=0$ has real roots greater than 1, show that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.

Let $a,\,b,\,c,\,d,\,e,\,f$ be real numbers such that the polynomial $P(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f$ factorizes into eight linear factors $x-x_i$ with $x_i>0$ for $i=1,\,2,\,\cdots,\,8$. Determine all possible values of $f$.

Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12$. Find $|f(0)|$.

I could simplify the expressions in the numerator and denominator to (1+x^n)/(1+x) as they are in geometric series and I used the geometric sum formula to reduce it. Now for what value of n will it be a polynomial? I do get the idea for some value of n the simplified numerator will contain the...

If this question is in the wrong forum please let me know where to go. For p, the vector space of polynomials to the form ax'2+bx+c. p(x), q(x)=p(-1) 1(-1)+p(0), q(0)+p(1) q(1), Assume that this is an inner product. Let W be the subspace spanned by . a) Describe the elements of b) Give a basis...

##x^2(3x^2+4x-12) +k=0## ##(3x^2+4x-12)= \frac{-k}{x^2}## or ##(4x^3-12x^2)=-k-3x^4## ##4(3x^2-x^3)=3x^4+k## ##4x^2(3-x)= 3x^4+k## or using turning points, let ##f(x)= 3x^4+4x^3-12x^2+k## it follows that, ##f'(x)=12x^3+12x^2-24x=0## ##12x(x^2+x-2)=0## ##12x(x-1)(x+2)=0## the turning points...

For an integer $n\ge 2$, find all real numbers $x$ for which the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value.

If a polynomial $P(x)=x^3+Ax^2+Bx+C$ has three real roots at least two of which are distinct, prove that $A^2+B^2+18C>0$.

Some questions: Why is this even a valid wave function? I thought that a wave function had to approach zero as x goes to +/- infinity in all of space. Unless all of space just means the bounds of the square well. Since we have no complex components. I am guessing that the ##\psi *=\psi##. If...

Proof: ##(\Leftarrow)## Suppose there exists non zero ##b \in R## such that ##bp(x) = 0##. Well, ##R \subset R[x]##, and so by definition of zero divisor, ##p(x)## is a zero divisor. (assuming ##p(x) \neq 0##). ##(\Rightarrow)## Suppose ##p(x)## is a zero divisor in ##R[x]##. Then we can choose...

This is my solution however I feel like the number is far too big can anyone see what I’ve done wrong

Hello everyone. I need to construct in python a function which returns the evaluation of a Chebishev polynomial of order k evaluated in x. I have tested the function chebval form these documents, but it doesn't provide what I look for, since I have tested the third one, 4t^3-3t and import numpy...

There is an arbitrarily complicated function F(x,y,z). I want to find a simpler surface function G(x,y,z) which approximates F(x,y,z) within a region close to the point (x0,y0,z0). Can I write a second-order accurate equation for G if I know F(x0,y0,z0) and can compute the derivatives at the...

Find the equation of a quartic polynomial whose graph is symmetric about the y -axis and has local maxima at (−2,0) and (2,0) and a y -intercept of -2

Help me if what I have done so far can be simplified further.

To find the coefficients of the Taylor polynomial of degree two of the function ##z(x,y)## around the point ##(0,0)##, what would be a handy way of doing that in python? How would one find the derivatives of ##z(x,y)##?

Summary: Worth teaching in secondary school? - or too bewildering? The mathologer video made me aware of Lill's method for solving polynomials with real roots. Although I'm not involved in secondary school teaching, I can't help wondering if it is a suitable topic for that level. Perhaps...

I can do question (a). For question (b), I can not see the relation to question (a). Can we really do question (b) using result from (a)? Please give me little hint to relate them Thanks

Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...

the polynomial f(x) = ax4 - 3x3- 63x2+ 152x - b has one of its zeros at x = 5 and passes through the point (-2, -560) Question: Use this info to find the values of a and b I am prepping for a test and this one question is really stumping me, I wondered if anyone would be able to help. For all...

Dear Everybody,I am having trouble with how to begin with this problem from Abstract Algebra by Dummit and Foote (2nd ed): Let $R$ be a commutative ring with 1. Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in...

Hello!(Blush) I learned about Eisenstein's irreducibility criterion. But it's still hard for me to implement it when the integer coefficients may be as larger as 1e9. What's more, how can I (or my computer?) know when to change x into x+a? It really puzzles me(O_o)??

i have some doubts from chapter 1 of the book Mathematical methods for physics and engineering. i have attached 2 screenshots to exactly point my doubts. in the first screenshot...could you tell me why exactly the 3 values of f(x) are equal. the first is the very definition of polynomials...but...

I was reading this book - " mathematical methods for physics and engineering" in it in chapter 1 its says "F(x) = A(x - α1)(x - α2) · · · (x - αr)," this makes sense to me but then it also said We next note that the condition f(αk) = 0 for k = 1, 2, . . . , r, could also be met if (1.8) were...

So, the values of polynomial ##p## on the complex unit circle can be written as ##\displaystyle p(e^{i\theta}) = a_0 + a_1 e^{i\theta} + a_2 e^{2i\theta} + \dots + a_n e^{ni\theta}##. (*) If I also write ##\displaystyle a_k = |a_k |e^{i\theta_k}##, then the complex phases of the RHS terms of...

I'm reading a book where the author gives the long division solution of ##\frac 1 {1+y^2}## as ##1-y^2+y^4-y^6...##. I'm having trouble duplicating this result and even online calculators such as Symbolab are not helpful. Can anyone explain how to get it?

x^2 - x -3 + 2c = 2x(ax+b) x^2 -2ax^2 - 2bx - x - 3 + 2c = 0 x^2(1-2a) -x(1+2b) -3 + 2c =0 Using girard r1+r2 = (1+ 2b)/(1-2a) r1xr2 = (-3 +2c)/(1-2a) After this I am stuck. Thank you.

At first I was thinking about using the dirac delta function ##\delta(x-1)##, but then I recalled ##\delta \notin L_2[0,1]##. Any ideas? I'm thinking no such function exists.

f(x) = A(x) . (x2 + 4) + 2x + 1 f(x) = B(x). (x2 + 6) + 6x - 1 f(x) = C(x) . (x2 + 6) . (x2 + 4) + s(x) Then I am stuck. What will be the next step? Thanks

Hi PF! I'm trying to solve the polynomial eigenvalue problem ##M \lambda^2 + \Phi \lambda + K## such that K = [5.92 -.9837;-0.3381 109.94]; I*[14.3 24.04;24.04 40.4]; M = [1 0;0 1]; [f lambda cond] = polyeig(M,Phi,K) I verify the output of the first eigenvalue via (M*lambda(1)^2 +...