This tag is for questions containing common root(s) between two or more polynomial equations.
Questions tagged [common-root]
29 questions
11
votes
4 answers
Find a such that $ax^{17}+bx^{16}+1$ is divisible by $x^2-x-1$.
Find $a$ such that $ax^{17}+bx^{16}+1$ is divisible by $x^2-x-1$.
I tried taking the roots of the polynomial which are $\frac{1±\sqrt{5}}{2}$
And I got the equation $a(\frac{1±\sqrt{5}}{2})^{17}+b(\frac{1±\sqrt{5}}{2})^{16}+1=0$
Now I don't know…
Toshu
- 1,109
- 14
- 30
6
votes
1 answer
On the joint numerical range of a pair of symmetric matrices
Proposition 13.4 of Alexander Barvinok's A Course in Convexity shows the existence of the following result:
Let $n\ge 3$. For two $n\times n$ symmetric matrices $A$ and $B$, and a PSD matrix $X$ with $\mbox{trace}(X) = 1$, there exists a unit…
Sam
- 323
6
votes
1 answer
Common roots of recursive defined polynomial
I have a series of polynomials $P_j(x)$ given by the recursive formula
$$P_{j+1}=\frac{e_j}{c_j}xP_{j}-\frac{f_j}{c_j}P_{j-1}
$$
with $P_{-1} \equiv 0$, $P_0 \equiv 1$, where
$$c_j = (j+1)(j+2\kappa+1),\\
e_j = (2j+2\kappa+1)(j+\kappa+1),\\
f_j =…
Mario
- 111
- 6
6
votes
3 answers
Common complex roots
If the equations $ax^2+bx+c=0$ and $x^3+3x^2+3x+2=0$ have two common roots then show that $a=b=c$.
My attempts:
Observing $-2$ is a root of $x^3+3x^2+3x+2=0\implies x^3+3x^2+3x+2=(x+2)(x^2+x+1)=0$
Hence $ax^2+bx+c=0$ can have complex roots in…
mnulb
- 3,701
4
votes
2 answers
If a quadratic equation can have real and equal roots, then why don't we say it has one root?
Suppose $ax^2+bx+c$ is a quadratic equation with $D=0$
So it has the roots $x=\frac{-b}{2a},\frac{-b}{2a}$ which are real and equal
Why don't we just say it has one root which would be $x=\frac{-b}{2a}$?
3
votes
2 answers
numerically solving a system of multivariate polynomials
I have a system of polynomial equations which I need to solve numerically. The one Im currently interested in has around 20 variables and a similar number of equations. All coefficients are rational numbers. In the end I am only interested in real…
Simon
- 5,161
3
votes
1 answer
Common root of cubic and quadratic equation
If equations $ax^3+2bx^2+3cx+4d=0$ and $ax^2+bx+c=0$ have a non zero common root, prove that $(c^2-2bd)(b^2-ac) \geq 0$.
I know the condition of common root of two quadratic equations but I have no idea on how to proceed with this question.
user600016
- 2,235
3
votes
1 answer
Common roots of irreducible polynomials
Is it possible for two distinct irreducible polynomials with integer coefficients to have a root in common? In other words, is it possible that a root is shared by some two distinct irreducible polynomials?
I would say no, because then it could be…
Ecir Hana
- 1,015
- 1
- 10
- 20
2
votes
2 answers
Two quadratic equations having a common root
If quadratic equations $a_1x^2+b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=0$ have both their roots common, then they satisfy
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$
But even if both the quadratic equations have only one common root(say $\alpha$),…
marks_404
- 677
2
votes
0 answers
Roots with same size for polynomials with distinct degrees
I am working on a problem which just arrived at the following question related to roots of polynomials.
Let $f(x)=x^7-2x^6+1$ and $g(x)=x^{10}-2x^9+1$. These polynomials have $x=1$ as their unique common root (because $f(x)/(x-1)$ and $g(x)/(x-1)$…
Jean
- 629
2
votes
0 answers
Find a polynomial $f \in \mathbb{Q}[x]$ with the least possible degree s.t. $\gcd(f,f')= x^5-5x^4+\frac{25}{4}x^3, f(1)=3$
We know that $f,f'$ have exactly the same roots their gcd has (and that $f'$ also matches their multiplicity). These are $x=0$ with multiplicity $3$ and $x=\frac{5}{2}$ with multiplicity $2$.
Therefore, accounting for the increment in multiplicity,…
ydnfmew
- 835
2
votes
2 answers
Is the following reasoning right?
I saw this post on Reddit and it had the following comment
√x²=|x|
It's the principle square root solution. So the square root of 49 is 7. It's the definition of square root.
However the equation x²=7² has two solutions +7 & -7
Only because…
penguin99
- 189
1
vote
1 answer
Common root of two equation
If the root of the equation 3x³+Px³+Qx-37=0 are each one more than the roots of the equation x³-Ax²+Bx-C, where A,B,C,P & Q are constants, then the value of A+B+C is equal to :
Actually I solved this question by let a root of 3x³+Px³+Qx-37=0 is t…
Soni
- 13
1
vote
1 answer
Solution verification on a question about polynomials and the midpoints of their intersection
Could someone check my solution to the following question?
The above image describes the curve $y=x-x^3$ and the line $f(x)=m(x-p)+p-p^3$ where $m$ is the gradient of the line and $p<-1$. The two curves intersect at points $P(p,p-p^3), Q$ and $R$.…
Pen and Paper
- 1,371
1
vote
1 answer
Why do we need Bezout's Theorem in this statement?
I'm analyzing a paper concerning algebraic geometry. In a proof, author uses Bezout's theorem. But I don't get it much. Here is the summary:
Let $K(t,s)$ and $F(t,s)$ be polynomials in $t,s$ with $F$ irreducible. The zero set of $K$ includes the…
ugurgozutok
- 21