Tagged Questions
3
votes
2answers
141 views
Does this polynomial factorise further?
I just did a national exam and this question was in it, I am convinced this does not work:
Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorise this cubic fully.
My attempt
1 | ...
1
vote
3answers
48 views
find out the value of $\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$
If $(x-3)^2+(y-5)^2+(z-4)^2=0$,then find out the value of $$\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$$
just give hint to start solution.
2
votes
0answers
31 views
Find the factorization of the polynomial as a product of irreducible [duplicate]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$
Testing with the simplest possible root in this case, $P(1)=0$
Applying the ...
5
votes
6answers
222 views
Cubing a simple thing
I am trying to expand $\quad (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. ...
4
votes
2answers
95 views
Irreducibility of $x^n-x-1$ over $\mathbb Q$
I want to prove that
$p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible.
My attempt.
GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
0
votes
1answer
21 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
80 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:
...
1
vote
3answers
66 views
Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$
For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$
where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
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? ...
2
votes
1answer
32 views
Creating and using calibration factors
Perhaps simple question, but I (the simple) need some guidance. The following applies to a project ongoing and is a challenge in that I am not a math whiz!
As example, I wish to measure temperature ...
7
votes
2answers
167 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: $$\left(x^4-\frac{x}{2}\right)^2+\frac{3}{4}x^2-x+1 = ...
2
votes
2answers
64 views
What is “prime factorisation” of polynomials?
I have the following question:
Find the prime factorisation in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreduciblity in $\mathbb{Z}[x]$, of ...
2
votes
2answers
47 views
Create a formula by given solutions
For my upcoming middle school exams I will need to convert a formula.
I have got the following question:
Create a formula which has the following solutions: $$ x_{1} = 5,\quad x_{2} = -3.$$
The ...
1
vote
2answers
57 views
Is $x^2+1$ irreducible over a cyclotomic field?
Let $K=\mathbb{Q}[\omega]$, where $1+\omega+\omega^2=0$, let $f(X)=X^2+1$. How can i prove irreducibility of $f$ over $K$?
5
votes
6answers
137 views
Show that $x^4 +1$ is reducible over $\mathbb{Z}_{11}[x]$ and splits over $\mathbb{Z}_{17}[x]$.
Reduction into linear factors $\mathbb{Z}_{17}[x]$:
This part is not too hard: $x^4 \equiv -1$ mod 17 has solutions: 2, 8, 9, 15 so
$(x-2)(x-8)(x-9)(x-15) = x^4 -34 x^3 +391 x^2-1734 x+2160 \equiv ...
3
votes
5answers
153 views
Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$?
Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$?
The question is mentioned in my math olympiad. Please explain how to solve the problem. I have ...
0
votes
2answers
71 views
How to factor a polynomial modulo p?
Is there a general strategy to factoring a polynomial modulo p? I've looked on Google but I've had a hard time finding anything that specifically outlines an approach that I can understand.
6
votes
4answers
187 views
Factorize $3m^4-6m^3+14m^2-6m+11$
I have this expression:
$3m^4-6m^3+14m^2-6m+11=0$ and I want to factorize it in $(m^2+1)(3m^2-6m+11)$.
How can I do it? Thanks for any help!
8
votes
1answer
135 views
How prove that polynomial has only real root.
Let this polynomial $f(x)=\displaystyle\sum_{i=1}^{n}a_{i}x^i,\;\;a_{i}\in \mathbb{R} $ have only real roots. Prove:
The polynomial $g(x)=\displaystyle\sum_{i}^{n}C_{n}^{i}a_{i}x^i$ has only real ...
3
votes
2answers
73 views
Write in polynomial in factored form in complex number
Write the following polynomial in factored form(in complex number):
$$1+z+z^2+z^3+z^4+z^5+z^6$$
Also, is there general solution of factoring for $1+z+z^2...z^n$ types of polynomial?
2
votes
3answers
79 views
Let $F$ be a field, then the polynomial $x^n - 1$ has $n$ roots in $F$ if $F$ contains a multiplicative subgroup of order $n$.
Is this following true? If $F$ is a field, then the polynomial $x^n - 1$ has $n$ roots in $F$ whenever $F$ contains a multiplicative subgroup of order $n$.
1
vote
5answers
103 views
How to factor a strange trinominal.
I know how to factor normal trinomials, however I was stumped when I saw this on my homework, could anyone help me through this?
The trinomial is as such:
$-m^2 + 8m + 18$.
2
votes
3answers
95 views
irreducibility of polynomials with integer coefficients
Consider the polynomial
$$p(x)=x^9+18x^8+132x^7+501x^6+1011x^5+933x^4+269x^3+906x^2+2529x+1733$$
Is there a way to prove irreducubility of $p(x)$ in $\mathbb{Q}[x]$ different from asking to PARI/GP?
0
votes
1answer
45 views
Factorising and limits
How do I factorize this expression? $$(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}}$$
...
4
votes
3answers
82 views
Factoring $x^8-x^4+1$ over $GF(7)$
Could anyone suggest any good way to do it? (The only way I can think of is by looking for roots (There are none), checking a factorization into the product of a 6 and a 2 polynomial (Many unknowns ...
7
votes
1answer
71 views
Factoring a couple $5$th degree polynomials
I'm reading an old (1895) textbook on algebra (doing a bit of review), and practicing factoring polynomials. The author started with polynomials where all terms share a common factor, like $4a^2 + 4a ...
1
vote
3answers
97 views
Trouble with factorising a polynomial
I'm supposed to show that:
$$y=\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$
and the required answers are: $$ P=5, Q=4, R=-4 $$
I tried to solve this with partial ...
0
votes
1answer
23 views
Distinct-degree factorization
I'm trying to understand distinct-degree factorization from Wikipedia.
I'm trying the algorithm on paper with $q=9$ and $f(x) = (x+4)(x+5) = x^2+2 \in F_{q}$.
We start with $i=1$. I calculate $g = ...
4
votes
5answers
91 views
Polynomial factoring $1-3x+4x^3$
I want to factorize (or factor ? can both verbs be used ?) $1-3x+4x^3$. I notice that $\frac{1}{2}$ and $-1$ are roots of the polynomial.
My questions are :
1) how do you notice that $\frac{1}{2}$ ...
1
vote
1answer
75 views
Can the product of irreducible polynomials have non-constant factors other than those polynomials?
Can the product of irreducible polynomials over the reals, $P_1, P_2,...,P_n$, have non-constant polynomial factors other than those polynomials or products of them (eg. $P_1P_3$)? It seems that the ...
2
votes
4answers
300 views
Factoring Polynomials with four terms and two variables
I've been working on this for hours and cannot figure it out.
When I search, I find factorization techniques that I already know but don't seem to be able to apply here, or that are for polynomials ...
1
vote
0answers
50 views
What is the condition for a polynomial to be factorizable in linear real factors?
I have a polynomial $p_a(x,y)= x^2F(a)+y^2G(a)-xH(a)-I(a)$ where $F(a)$, $G(a)$, $H(a)$ and $I(a)$ some real fuctions of $a$ are. Which conditions must satisfy $a$ so that I can factorize the ...
1
vote
1answer
93 views
Factoring a polynomial with big integer coefficients and some known factors.
I have the following polynomial that I want to factor
$$
\begin{align*}
p(x)=
&- 236364091 x^{13}- 28363690920 x^{12}- 1487737229594 x^{11}\\
&- 44880832661940 x^{10} - 860924276925225 x^9- ...
16
votes
1answer
280 views
Irreducibility of $x^{n}+x+1$
Motivated by this problem, and KCd's comment on my answer, I am left with the following question:
Question: Suppose that $n\not \equiv 2\pmod{3}$. Is $$x^n+x+1$$ irreducible over $\mathbb{Q}$?
...
0
votes
3answers
55 views
1
vote
1answer
34 views
Polynomial factoring issue
I am dealing with an issue for which I do not find answer on the Internet.
When I factorize a polynomial, I can get this structure:
$$
(x-a)(x-b)(x-c)^2
$$
But sometimes I have seen others like:
$$
...
6
votes
5answers
608 views
Is $x^4+4$ an irreducible polynomial?
We know that $p(x)=x^4-4=(x^2-2)(x^2+2)$ is reducible over $\mathbb{Q}$ even not having roots there.
What about $q(x)=x^4+4\in \mathbb{Q}[x]$? Again, no roots.
1
vote
2answers
233 views
Factoring polynomials of degree 6 in 2 ways.
Let $P(x)$ be an integer polynomial of degree $6$ that is irreducible over the integers.
$P(x) = x^6 + (A+a) x^5 + (B+ aA+ b) x^4 + (C+aB+bA +c) x^3 + (aC +bB +cA) x^2 + (bC+cB) x + cC = x^6 + ...
1
vote
1answer
111 views
Factoring polynomials of degree 6 over extension fields.
Let $f(x)$ be a polynomial with integer coefficients that is irreducible over the integers and has degree 6.
Let $L$ be the splitting field of $F$. Then we can ask, whether there exist intermediate ...
3
votes
2answers
141 views
Factoring in Z3[x]
I need to factor $x^6+x^4+x^2+1$ into irreducible parts in $Z_3[x]$. Obviously this polynomial reduces to $(x^4+1)(x^2+1)$ which is irreducible in $Z[x]$, but I'm not sure how to confirm that it's ...
0
votes
5answers
68 views
Factorise $f(x)=x^5-x$ into a product of irreducibles in $\mathbb Z_{5}[x]$
So plugging in $1$ gives $f(1) = 0$ which means $1$ is a root and $f$ has a factor $(x-1)$ which is $\equiv (x+4)$ in $\mathbb Z_{5}[x]$ ?
I then divide $f(x)$ by $(x+4)$ using polynomial long ...
0
votes
4answers
71 views
Factor $x^3-x^2+2$ in $\mathbb Z_{3}[x]$
Factor $x^3-x^2+2$ in $\mathbb Z_{3}[x]$ and explain why the factors are irreducible.
So the factor is supposed to be:
$x^3-x^2+2 = 2(x + 1)(2x^2 + 2x + 1)= (x + 1)(x^2 + x + 2)$.
But I don't ...
4
votes
2answers
140 views
Solve logarithmic equation
I'm getting stuck trying to solve this logarithmic equation:
$$
\log( \sqrt{4-x} ) - \log( \sqrt{x+3} ) = \log(x)
$$
I understand that the first and second terms can be combined & the logarithms ...
0
votes
3answers
111 views
Reducible polynomial + integer = Reducible polynomial?
Reducible polynomial + integer = Reducible polynomial ?
As the title says.
More specific :
For every integer $n$, does there exist a pair of polynomials $p(x)$ and $q(x)$ such that:
...
0
votes
0answers
95 views
Factoring polynomials $f(g(x))$ over extension fields.
This question is a variation on another one :
related question
Let $f(x)$ and $g(x)$ be polynomials with integer coefficients and discriminant of $f(x)$ > discriminant of $g(x)$ , which are ...
1
vote
2answers
495 views
Graphing Cubic Functions
I'm having a Little bit of trouble in Cubic Functions, especially when i need to graph the Turning Point, Y-intercepts, X-intercepts etc. My class teacher had told us to use Gradient Method:
lets ...
1
vote
1answer
101 views
Factoring polynomials of degree $a p^b$ over extension fields.
Let $f(x)$ be an irreducible polynomial with integer coefficients, which is irreducible over $\mathbb{Z}$ and has degree $a p^b > p$ with $p,a,b>0$ and $p$ a prime.
It appears that $f(x)$ ...
0
votes
1answer
48 views
Distinct-degree factorization in finite fields
In $\mathbb{Z}_3$
$$ x^9 : x^4+x^3+x^2+2x+1 = x^5+2x^4+2x^2+2x$$
with remainder of $x$.
In $\mathbb{Z}_7$
$$x^7 : x^4+5x^3+x+5 = x^3+2x^2+4x$$
with remainder of $x$.
Is this random? Or is there ...
3
votes
1answer
118 views
Roots of rational equation with multiple variables?
Let's say we have a rational polynomial in $k$ variables. We are only interested in rational solutions. If $k = 1$, name the variables ${x}$, if $k = 2$, name them ${x,y}$.
For $k = 1$, it can be ...
7
votes
2answers
526 views
Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?
We can solve (get some kind of answer) equations like:
$$ ax^2 + bx + c=0$$
$$ax^3 + bx^2 + cx + d=0$$
$$ax^4 + bx^3 + cx^2 + dx + e=0$$
But why is there no formula for an equation like $$ax^5 + ...
