Tagged Questions
2
votes
1answer
19 views
For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
4
votes
2answers
57 views
Simplifying this expression $(e^u-1)(e^u-e^l)$
Is it possible to write the following
$$(e^u-1)(e^u-e^l)$$
as
$$e^{f(u,l)}-1?$$
5
votes
6answers
207 views
Cubing a simple thing
I am trying to do
$$(x + 2)^3 $$
I am actually not to sure what to do from here, the rules are confusing. To square something is simple, you just foil it. It is easy to memorize and execute. Here ...
0
votes
1answer
20 views
Relationship between 2 Dimensional Quadratic systems and roots
Given four points
$(x_1, y_1)
(x_2, y_2)
(x_3, y_3)
(x_4, y_4)$
How does one construct a system of two equations:
$a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$
$b_1x + b_2x^2 + b_3y + b_4y^2 + ...
9
votes
5answers
77 views
Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$
Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed:
...
4
votes
5answers
156 views
Checking whether a polynomial of high degree is bijective or not.
Let $P(x)$ be a polynomial of degree $101$. Then $x\mapsto P(x)$ cannot be a one-one onto mapping, i.e., bijective function from $\Bbb{R}$ to $\Bbb{R}$. True or false?
I think is when we take ...
2
votes
4answers
135 views
Interception with $x$-axis - not so trivial?
I want to find the interception with the x-axis of the following function:
$f(x) = \frac{1}{4}x^4-x^3+2x$.
So putting $0 = \frac{1}{4}x^4-x^3+2x$ I would get $0 = x(\frac{1}{4}x^3-x^2+2)$ but how to ...
1
vote
1answer
53 views
Finding the value of polynomial at a particular value / constant is given
If $P(k)$ is a polynomial of degree $8$ and $P(k) = \frac{1}{k}$ for $k = 1,2 ,3,\ldots,9$ then find the value of $P(10)$.
As we know the following :
$f(x) =a_nx^n+a_{n-1}x^{n-1}+.....a_1x+a_0$
...
1
vote
2answers
57 views
How to prove an odd-degree polynomial starts and ends at values of different sign?
Consider an odd-degree polynomial. How to prove that it starts a value that has different sign from its end value?
Or
$$
\lim_{x\to -\infty} f(x) \lim_{x\to+\infty } f(x)<0
$$
Please don't use ...
0
votes
2answers
45 views
Solving cubic equations
I was trying to solve a cubic equation which is :
$ -\lambda³ -\lambda² + 10 \lambda - 8 = 0$
I googled about it and I found the Rational Root theorem which is takes time to do it, but I found that ...
1
vote
2answers
106 views
Condition for fourth degree polynomial to have all real roots
For what range of values of $a$ will the following fourth degree polynomial have all real roots:
$$x^4 - 2ax^2 + x + a^2 -a = 0$$
3
votes
4answers
113 views
Solving this 3-degree polynomial
I'm trying to factor the following polynomial by hand:
$-x^3 + 9x^2 - 24x + 20 = 0$
The simplest I could get is:
$-x^2(x-9) - 4(5x+5) = 0$
Any ideas on how I could go ahead and solve this by hand? ...
1
vote
2answers
22 views
Find real solutions of polynomial
How with you find the solutions to the following:
$\sqrt{3x+10}-\sqrt{x+2}=2$
This is what I tried so far:
$(\sqrt{3x+10}-\sqrt{x+2})^2=2$
$(3x+10)+(x+2)-2\sqrt{(x+2)(3x+10)}=2$
...
4
votes
3answers
76 views
Prove a polynomial has all roots different
I need to prove that $P(x)=x^4+\zeta x+1$ where $\zeta\in\mathbb{R}$ and $\zeta\neq0$ has four different roots.
I have tried with the rule of signs of Decartes but it does not give enough information.
...
1
vote
2answers
44 views
Finding the Cube Root of a Polynomial
Question: Let $P = 27+108x+90x^2-80x^3-60x^4+48x^5-8x^6$, find $\sqrt[3]{P}$.
Just wondering if there is a general way for dealing with this sort of question, I was able to figure how to find ...
2
votes
4answers
63 views
Show that $1$ and $2$ are zeros of the following polynomial
Show that $1$ and $2$ are zeros of the polynomial $P(x)=x^4-2x^3+5x^2-16x+12$ and
hence that $(x-1)(x-2)$ is a factor of $P(x)$
0
votes
1answer
39 views
Finding a function which fits this data?
I need to find a polynomial (or other continuous elementary function) on the interval [70, 180] such that it passes through the points (70, 0) (this is a relative min), (105, 17) (this is a relative ...
7
votes
2answers
165 views
Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$
Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$
My try: $$\left(x^4-\frac{x}{2}\right)^2+\frac{3}{4}x^2-x+1 = ...
0
votes
0answers
47 views
Polynomial-Why is the answer for removing parentheses and solving different
I am a 45 yo self-taught software engineer and I am finally studying mathematics. Pretty simple question:
I am using the ALEKS software and during two different lessons on polynomials we get ...
2
votes
2answers
47 views
Create a formula by given solutions
For my upcoming middle school exams I will need to convert a formula.
I have got the following question:
Create a formula which has the following solutions: $$ x_{1} = 5,\quad x_{2} = -3.$$
The ...
0
votes
0answers
58 views
Finding the number of negative and the positive zeros
$$f(x)=x^{4} -3x^{3}+5x^{2}-x-2$$
How to find the number of negative and the positive zeros?
2
votes
3answers
31 views
If I have a polynom $p$ with $p(a) = 0$, how to construct a polynom $q$ with $q(a^{-1}) = 0$.
If I have a polynom $p$ which has a field element a as its root, i.e. $p(a) = 0$, how can I construct a polynom $q$ from it with $q(a^{-1}) = 0$. I conjecture that Vieta's formulae might be helpful, ...
2
votes
3answers
94 views
How to expand $(a_0+a_1x+a_2x^2+…a_nx^n)^2$?
I know you can easily expand $(x+y)^n$ using the binomial expansion. However, is there a simple summation formula for the following expansion?
$$(a_0+a_1x+a_2x^2+...+a_nx^n)^2$$
I found something ...
3
votes
5answers
152 views
Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$?
Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$?
The question is mentioned in my math olympiad. Please explain how to solve the problem. I have ...
1
vote
2answers
48 views
Larger Theory for root formula
Consider the quadratic equation:
$$ax^2 + bx + c = 0$$
and the linear equation:
$$bx + c = 0$$.
We note the solution of the linear equation is
$$x = -\frac{c}{b}.$$
We note the solution of the ...
2
votes
1answer
21 views
Constructing a specific polynomial?
I need to construct a polynomial with zeroes at $3$, $5$, and $10$ (and the function can't just be tangent at those points, it has to go below/above the axis. Also, there can't be any zeroes in ...
1
vote
5answers
53 views
Every polynomial of degree $\le m$ can be written in a form using its $m+1$ values
I'm not sure if the title is comprehensible. What I mean is this:
I've found here the following corollary:
Let $x_1, ..., x_{m+1}$ be $m+1$ different points in $\mathbb{R}$, and for $i = ...
9
votes
2answers
256 views
1
vote
1answer
41 views
Algebraic functions
I am wondering, if you consider a polynomial in two variables like
$$P(x,y)=0,$$
and a zero $P(a,b)=0$ exists fulfilling $P_y(a,b)=0$ and $P_x(a,b)=0$, is there continuity in the sense of
$$\forall ...
0
votes
2answers
41 views
number of roots of polynomial of order n
from theorem of algebra,it is well know that polynomial of order n has exactly n roots,for exmaple quadratic equation like $ax^2+bx+c$ has three cases
let $D=b^2-4ac$ ,so we have
...
0
votes
1answer
45 views
Factorising and limits
How do I factorize this expression? $$(2^n-3^n+n4^n)^{\frac{1}{n}}$$ so far I have: $$n4^n\left(\frac{1}{n} \left(\frac{1}{2}\right)^n-\frac{1}{n}\left(\frac{3}{4}\right)^n +1\right)^{\frac{1}{n}}$$
...
2
votes
1answer
58 views
Conclusion about Zeros of a polynomial ,when sum of it's coefficients is zero
I have a polynomial of the form:
$$\sum_{m=0}^k\frac{(-1)^{m+1}(4k-2m)!x^{2k-2m}}{m!(2k-m)!(2k-2m+1)!}$$
or identically:
$$\sum_{m=0}^k\frac{(-1)^{m+1}(4k-2m)!(x^{2})^{k-m}}{m!(2k-m)!(2k-2m+1)!}$$
...
1
vote
2answers
71 views
Polynomial roots
How is it possible to construct two distinct third degree polynomial equations with real coefficients and roots $2$ and $2+ i$?
Isn't the only possibility $p(x)=(x-2)(x-2-i)(x-2+i)=0$? Am I losing ...
0
votes
2answers
65 views
$f(x) = ax^5-17x^4-15x^3+153x^2-122x-b$, solve for $ a, b$
The polynomial function $f(x) = ax^5-17x^4-15x^3+153x^2-122x-b$ has one of its zeroes at $x=5$ and passes through the point $ (1,-64).$
a) Find the values of $a$ and $b$
b) Determine all the factors ...
3
votes
3answers
120 views
How to find all the solutions to $-8x^4-18x^2-11+\frac{1}{x^4}=0$?
How to know all the roots for fourth degree equation like this :
$$-8x^4-18x^2-11+\frac{1}{x^4}=0$$
1
vote
2answers
121 views
Division algorithm for multivariate polynomials?
We know that if $F$ is a field and $f(X)$ a non-zero polynomial in $F[X]$, then for every polynomial $g(X)$ we can find $q(X),r(X)$ such that
$$g(X)=f(X)\cdot q(X)+r(X)$$
with $r(X)$ the zero ...
2
votes
6answers
173 views
Polynomials with integer coefficients
Through definitions, theorems and my professor the following is true:
The product of any two odd integers is odd.
The sum and difference of any two odd integers are even.
The sum, product and ...
0
votes
2answers
91 views
Root of a polynomial with rational coefficients
I am currently learning about Direct Proofs. I am struggling trying to find a starting point to prove the Statement: For all real numbers $c$, if $c$ is a root of a polynomial with rational ...
10
votes
0answers
91 views
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes.
I've been thinking about this question, but ...
2
votes
1answer
45 views
Second solution for search of negative roots
How many $\underline{\text{negative roots}}$ does the equation $x^4-5x^3-4x^2-7x+4=0$ have?
My reasoning:
I rewrote the equation like: $$x^4-5x^3-4x^2-7x+4=0 \Rightarrow (x^2-2)^2 = 5x^3+7x$$
For ...
3
votes
4answers
115 views
polynomial division problem ( find the remainder)
How to find out what remainder will $\,(x-1)^{2013}\,$ have upon division by $\,x^2-2x+2\,?\;$
I've never solved anything like this before, so I have no ideas at all..
Thanks in advance!
2
votes
1answer
105 views
Finding complex roots of a polynomial
I'm having trouble figuring out to find the complex roots of polynomials with degree greater than 2.
I particular, how would I find the (complex) roots of the following: $$x^4 + 5x^2 +4?$$ I know the ...
2
votes
4answers
291 views
Factoring Polynomials with four terms and two variables
I've been working on this for hours and cannot figure it out.
When I search, I find factorization techniques that I already know but don't seem to be able to apply here, or that are for polynomials ...
3
votes
0answers
60 views
Polynomial bound
Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that
$$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$
Suppose that $P(x)> 0$ for all ...
2
votes
5answers
84 views
Is it more practice and intuition or rather algorithmic to solve third degree polynomials of this type?
Consider
$x^3 - 6x^2 + 11x - 6 = 0$
I can not reasonably factor this intuitively in any short amount of time with my skill level. Is this the only hope to solving such equations by hand? What tools ...
3
votes
2answers
126 views
Synthetic division via the greedy strategy
I was looking at the expanded synthetic division within Wikipedia. I was stumped by how to come up with and perform the 'compactified' version of synthetic division.
Does anyone know how to do it?
4
votes
2answers
73 views
$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$
Given that the equation $$p(x)=a_0x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ has $n$ distinct positive roots, prove that
$$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$$
I had ...
4
votes
1answer
78 views
How to find $1/x^3 + 1/y^3$?
If I am given, $x + y = a$ and $xy = b$, how would I find the value of $\dfrac1{x^3} + \dfrac1{y^3}$?
2
votes
1answer
65 views
Query about cubic roots and the discriminant
I have an equation a little like this;
$$2c^3 - 3oc^2 + oG = 0$$
In my equations, $o$ is always $> G$ . I'm trying to find a general solution rather than constantly estimating it graphically, ...
2
votes
2answers
88 views
How to multiply polynominals
I can't figure out how to multiply these polynominals $$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$$
I tried multiplying like this
$$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$$
...
