What is Polynomial: Definition and 1000 Discussions
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.
Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.
i understand how to find minimal poly. if a matrix is given. i am curious if you can find the matrix representation if minimal polynomial is given.
i'm not exactly sure how you could since you can possibly lose repeated e-values when you write minimal polynomial. how can u create a n...
I'm doing a calculation which finds the characteristic polynomial of a matrix, HH, with rather complex entries and then determines the discriminant of that polynomial. For smaller matrices up to around 7x7 it finishes evaluating the Discriminant command within a few hours, but at a 10x10, which...
hi i have this homework question and I am not sure if my thought process is valid.
The Question:
let a, b and c be roots of the polynomial equation: x^3+px+q=0 and S(n)=(a^n)+(b^n)+(c^n)
now prove: that for S(n)= -p(S(n-2))-q(S(n-3)) for n>3my attempt:
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first off...
Homework Statement
z^3 + (-5+2i)z^2 + (11-5i)z -10+2i =0 has a real root, find all the solutions to this equation.
The Attempt at a Solution
I have only solved imaginary number polynomials with a given root, but this has no given root, how do I find the real solution? that I can then...
The polynomial z^4 + 2z^3 + 9z^2 - 52z + 200 = 0 has a root z=-3+4i. Find the other 3 roots.
Since the given root is complex, one of the other roots must be the complex conjugate of the given root. So the 2nd root is z=-3-4i. To find the other roots, I divided the polynomial by z^2 + 6z +...
Homework Statement
Solve the following equation:
x^4 + 12x^3 + 46x^2 + 60x + 20 = 0
Homework Equations
Well, I know how to solve simpler equations, in which the unknown dosen't appear at a power higher than 3. I tried to factor this polynom but I didin't suceed.
The Attempt...
Hi,
To calculate the intersection of two straight lines the cross product of the line vectors can be used, i.e. when the lines start in points p and q, and have direction vectors r and s, then if the cross product r x s is nonzero, the intersection point is q+us, and can be found from...
Homework Statement
Let t_j=j/100, a_j=j, b_j=-j, for j=0,1,...,99. Define f(t)=\sum\limits_{k=0}^{99} (a_k\cos(2\pi kt)+b_k\sin(2\pi kt))
Determine the values of c_l, d_m for l= 0,...5, m=1,...,4, so that P(t)=\frac{c_0}{2}+\sum\limits_{k=1}^4 (c_k\cos(2\pi kt)+d_k\sin(2\pi kt))+c_5\cos(10\pi...
Homework Statement
So, we are supposed to find the tenth taylor polynomial of sinx. On wolfram, I get a final term with x^11 at the end. How does that make sense?! According to the formula, n=10 so the maximum degree should be 10...
Homework Equations
Taylor polynomial formula.
The...
I am reading R.Y. Sharp: Steps in Commutative Algebra.
Lemma 1.13 on page 7 (see attachment) reads as follows:
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1.13 LEMMA. let R be a commutative ring, and let X be an indeterminate; let T be a commutative...
Homework Statement
S = {1, x, x^2}
Find ||1||, ||x||, and ||x^2||.Homework Equations
##\sqrt{v \cdot v}##The Attempt at a Solution
I don't know the components of each vector, so how can I perform the dot product?
Hello!
Tchebysheff polynomials are often defined with trigonometric functions:
T_m (x) =
\begin{cases} \cos(m \arccos (x)) & -1 \le x \le 1\\
\mathrm{cosh} (m \mathrm{arccosh} (x)) & x > 1\\(-1)^m \mathrm{cosh} (m \mathrm{arccosh} |x|) & x < 1
\end{cases}
But they are also polynomials, and...
I have given polynomial: x^3-8x^2+19x-12
I know how to find the roots with Horners method,i am just wondering if there is an easier and quicker way to find them? Thank you!
I think the solution to the radial schrodinger equation includes a form of the Laguerre polynomials, the polynomial v(ρ). Does anyone know what this v(ρ) polynomial is called? The only information my book gives is: "The polynomial v(ρ) is a function well known to applied mathematicians."...
Hi guys! I'm kind of stuck in my review here in Quantitave on the Polynomial part :confused:
Homework Statement
A. The first problem I had is Problem 7 on the link I will provide which has this.
(x + y)3 + (x-y)3 = ?
B. The second problem I had is Problem 8 on the same link I will...
I wonder what are the tecniques,or what is the easiest way to simplify given polynomial:
x^3-9x^2+27x-27
If possible,without Horners algorithm. Thank you!
Recently, I've developed a habit of trying to separate the idea of a function from the idea of the image of the function. This has mostly just confused me, but I am adamant about sticking to it.
I think the two terms, "ring of polynomials" and "ring of polynomial functions," are not...
why does a nth degree polynomial has atleast one root and a maximum of n root...?
In my book it's given, it's the fundamental theorem of algebra.
Is there a proof...?
Thank's for help. (In advance)
Hi,
I'm looking for a program that will take
[1 + (1+p)]*p
and return the unchanged polynomial. i.e. 2p + p^2
I know that MATLAB allows u to do polynomial convolution but for my purposes a program that returns the polynomial in the above format will be much much easier to work with...
$\displaystyle x^4+2x^3-8x^2+18x+9$
this is what i tried,
$x^4+2x^3-8x^2 = (x^2+4x)(x^2-2x)$
then,
$a(x^2+4x)+b(x^2-2x)=18x$
where ab=9
did i set up my solution correctly? can you tell me where I'm wrong.
Homework Statement
Show that if ##f:\mathbb{R}\to \mathbb{R}## is a polynomial function of odd degree, then ##f(\mathbb{R}) = \mathbb{R}##.
The attempt at a solution
How can I rigorously prove this? What is the most direct and concise way to prove this?
What I have is the following...
Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$.
Given that $f(4)=2009$, find $f(1)$.
Homework Statement
Homework Equations
The Attempt at a Solution
How did they go from the first step in the blue to the second step in the blue?
I tried integration by parts but that didn't work.
I have to find all solutions for X when:
x4-2x3-25x2+50x
I have done it so,but I am not sure if this is ok:
x(x3-2x2-25x+50)
= x(x2(x-2)-25(x+2)
= x(x2-25)(x-2)
=x(x-5)(x+5)(x-2)
Now i see that root/zeroes are +5,-5 and 2. I know that this polynomial has another zero that is 0,but how do i...
Homework Statement
A car is moving with speed 10m/s and acceleration 1 m/s^2 at the given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second.
Homework Equations
The Attempt at a Solution
I don't get how you're supposed to apply...
Hello guys,
I'd like to ask you how to efficiently factorize complicated polynomial like this one for example:
$$\frac{128t^4+128t^3+192t^2+32t+40}{(4t^2+1)^4}$$
I've spend more than a hour trying to decrypt and decompose the polynomial, but to no avail. For simple cubic polynomial I know...
Hi guys, somehow after a couple years of not doing math I got a bit rusty...
How do I solve x^3+3x^2-4=0 ?
I'm kinda stuck? I figured factorizing but I can't seem to find any good factors :/
Homework Statement
Was given a matrix
To find the eigenvalues I set up the characteristic equation
[-1-x | 7 | -5 ]
[-4 | 11-x | -6 ]
[-4 | 8 | -3-x]
With some dirty work I got this bad boy out, which I'm having trouble factoring
-x3+7x2-15x+9Homework Equations...
Hi everyone, :)
This is one of the thoughts that I got after thinking about finding the minimal polynomial of a matrix. I know that the minimal polynomial is easy to find when a matrix is diagonalizable. Then the minimal polynomial only consist all the distinct linear factors of the...
Hi everyone, :)
Here's another question that I solved. Let me know if you see any mistakes or if you have any other comments. Thanks very much. :)
Problem:
Prove that the eigenvector \(v\) of \(f:V\rightarrow V\) over a field \(F\), with eigenvalue \(\lambda\), is an eigenvector of \(P(f)\)...
this stunned me. i stare at these problem for a while and tried my best to factor it but no success. please help me solve this.
1. $x^3+x^2y^2+x^2+xy+y^3+4y-3xy^2-12x$
2. $3x^2+7xy-3xz-2yz+4y^2-6z^2$
can you guys help me factor this polynomial.
$\displaystyle 2x^2-4xy+2y^2+5x-3-5y$
$6x^2-xy+23x-2y^2-6y+20$
by the way this is what i tried in prob 1
$2(x-y)^2+5(x-y)-3$ -->> I'm stuck here. and in prob 2 i have no idea where and how to start.
thanks!
If I could use any polynomial up to degree ∞, then can I get a close fit to any continuous function?
I know that with a 4th degree polynomial you can get a pretty close fit to the sine function between 0 and 2pi...
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
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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...