0
votes
2answers
22 views

Polynomial identity for a sum

If $$f(x) = \sum_{i=0}^{n}A_i x^i \quad \text{ and } \quad g(x) = \sum_{i=0}^{n}B_i x^i$$ are two degree $n$ polynomials, then we can say that the polynomial $$h(x) = \sum_{k=0}^{2n}C_k x^k \quad ...
3
votes
1answer
53 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
0
votes
1answer
20 views

Show that a Polynomial has certain factorization

$P(x)$ is a polynomial in $x$ of degree $\leq n-1$. Show that $P(x)$ has $n-1$ distinct roots and thus has the factorization $$k\Pi_{i=2}^n(x-a_i)$$, where the constant $k$ is the coefficient of ...
0
votes
1answer
20 views

How to simplifying and solving this polynomial?

I have a problem with simplifying the polynomial. In the first time, I see that this polynomial is quite simple, but when I'm trying, I realized that this polynomial isn't as easy as I saw. Here is ...
5
votes
3answers
93 views

Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$

I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right? If I can use ...
2
votes
1answer
34 views

Factorising a complex polynomial over C

If $f(z)=z^3+7z^2+16z+10$, find all factors of $f(z)$ over $C$. If I had at least one zero or factor I would be able to find the others, but I just don't know how to start.
2
votes
2answers
54 views

Show that $P(X) -X$ divides $P(P(X))-X$

Let $P$ be a polynomial in $R[X]$. Then show that $P(X) -X$ divides $P(P(X))-X$
5
votes
3answers
213 views

Factor $x^4+1$ over $\mathbb{R}$

Factor $x^4+1$ over $\mathbb{R}$ Well, I read this question first wrongly, because the reader is about complex analysis, I did it for $\mathbb{C}$ first. I got. $x^4+1=(x-e^{\pi i/4 })(x-e^{3 ...
2
votes
4answers
106 views

Factoring $x^3 - x^2 - x + 1$

I'm familiar with basic forms of polynomial factoring like completing the square or factoring (e.g. finding that $x^2+x-6=(x+3)(x-2)$, but I'm currently working on integration of rational functions by ...
0
votes
2answers
50 views

How to factor cubics having no rational roots

$$-8x^3 +8x -3 = 0$$ I've already tried the possible roots of $\pm 1$ and $3$ using the rational roots test, but none of these help break it down into something more workable. How do I solve this ...
1
vote
2answers
53 views

If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
2
votes
2answers
49 views

Factor 9 terms with 3 variables into 4 expression

I just got the determinant from a 4x4 matrix and the simplified version is below. $$ det(M) = \begin{vmatrix} 2k-mw^2 & -k & 0 & 0 \\ -k & 2k-mw^2 & -k & 0 \\ 0 & -k ...
3
votes
3answers
128 views

Solve $t^4+4 t^3+6 t^2+4 t-32 t^{1/4}+1 = -16 $

I'm trying to solve the following equation: $$(t+1)^4 - 32 t^{\frac{1}{4}}=-16 $$ where t $\geq 0$, which is equivalent to $$t^4+4 t^3+6 t^2+4 t-32 t^{\frac{1}{4}}+1 = -16 $$ Wolfram Alpha tells that ...
0
votes
2answers
58 views

Factorization of a polynomial

I need to find the roots of this polynomial $$2x^2-x^4-x=0.$$ I know that it is necessary the factorization to obtain $$-x(x-1)(x^2+x-1)=0.$$ I asked to factorize my polynomial to Mathematica. The ...
0
votes
1answer
56 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
0
votes
4answers
41 views

Recognizing the proper polynomial factorization to solve an indeterminate limit

I had to solve the $\lim_{x \to 3} \frac{x^3-3x^2-x+3}{x^2-x-6}$ that is indeed an indeterminate form ($\frac{0}{0}$). The approach I adopted was to factor the polinomials so that I can deviate from ...
3
votes
1answer
65 views

Factoring $(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$

How to prove the following equality? $$(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$$ I did it $$\begin{aligned} a^2b + a^2c + ab^2 + cb^2 + bc^2 + ac^2 + 2abc &=a^2(b + c) + bc(b + c) + a(b^2 + ...
1
vote
6answers
151 views

How to factorize $2x^2+5x+3$?

I'm doing pre-calculus course at coursera.org and I'm in trouble with this solution $$2x^2 +5x +3 = (2x+3)(x+1)$$ By trial, using ac-method I got stuck: $$ ac = (2)(3) = 6\\ 6 + ? = 5 \Rightarrow~ ? ...
0
votes
1answer
30 views
1
vote
2answers
54 views

Question about Polynomial Factor Theorem

I was reading the solution to an algebra problem but got stuck at one part. Problem is here: (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) Example 4.2.6 -- page 140 of the PDF (the book ...
0
votes
1answer
60 views

$gcd(a,b)$ in a UFD subring is not a greatest common divisor in the ring

Give a counterexample that $R$ is a unique factorization domain but not a principal ideal domain, $S$ is a ring containing $R$, such that $a,b\in R$, $gcd(a,b)$ in $R$ is not a greatest common ...
1
vote
0answers
51 views

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$?

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$ ? For example: $1.$ $a^2+b^2+c^2$ $2.$ $a^2+b^2-c^2$ $3.$ $a^2+b^2+c^2+d^2$ $4.$ ...
0
votes
1answer
55 views

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain?

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain? I have no idea to prove or disprove this... should I prove ...
4
votes
4answers
76 views

$\omega^2+\omega+1$divides a polynomial

The question is Show that $f(n)=n^5+n^4+1$ is not prime for $n>4$. The solution is given as Let $\omega$ be the third root of unity. Then $\omega^2+\omega+1=0$. Since ...
2
votes
2answers
31 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
1
vote
1answer
33 views

Factors of $x^n+1$ over $\mathbb{Z}[x]$

Is there any equivalent to $x^n-1 = \prod\limits_{d|n} \phi_d$ where $\phi_d$ is the $d$th cyclotomic polynomial but for $x^n+1$? Even better, can we generalize any further?
-3
votes
2answers
71 views

How to factor $x^3-x^2+x+3$ [closed]

I have a partial fractions integral with $x^3-x^2+x+3$ in the denominator. How do I factor this?
0
votes
1answer
38 views

Inversion in factor rings

I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$ Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ? I should found ...
0
votes
2answers
98 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
2
votes
1answer
49 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
2
votes
1answer
149 views

Factorizing $(x-1)(x-3)(x-5)(x-7)-64$

We need to factorize: $$(x-1)(x-3)(x-5)(x-7)-64$$ We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$. So this means that if the ...
0
votes
0answers
33 views

Polynomial factorization over an infinite field - is there an algorithm

In my previous questioned I asked how do I factor a polynomial, and I gave an easy example of a polynomial of degree 2. But now I have another question I need to solve. I need to factor ...
3
votes
1answer
46 views

Polynomial factorization to irreducible factors with respect to field

I have a question, I think I don't understand this material very well and could use an explanation / some help. Basically we are asked to decompose $x^5-x$ to irreducible factors over $R,F2,F5,C$ ...
0
votes
1answer
33 views

$\frac{x^4 - x^3 + ax^2 + bx + c}{x^3 + 2x^2 - 3x + 1}$, remainder $3x^2 - 2x + 1$. Find $(a + b)c$.

Given the polynomials $P(x) = x^4 - x^3 + ax^2 + bx + c\\ Q(x) = x^3 + 2x^2 - 3x + 1\\ R(x) = 3x^2 - 2x + 1$ such that $P(x) = D(x)Q(x) + R(x)$, find $(a + b)c$. I would normally apply little ...
3
votes
1answer
64 views

Finding $a_n$ such that $x^n+a_1x^{n-1}+\cdots+a_{n-1}+a_n$ cannot be factored when $a_1,\cdots,a_{n-1}$ given

Let $n\ge 4\in\mathbb N$. Suppose that $a_1,a_2,\cdots,a_{n-1}$ are given integers. Then, here is my question. Question : Is the following true for any $(a_1,a_2,\cdots,a_{n-1})$ ? There ...
7
votes
2answers
110 views

Defining irreducible polynomials recursively: how far can we go?

Fix $n\in\mathbb N$ and a starting polynomial $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_n\ne0$. Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+a_rx^r$ such ...
1
vote
3answers
67 views

How do you factor $(10x+24)^2-x^4$?

I tried expanding then decomposition but couldn't find a common factor between two terms
1
vote
2answers
160 views

Getting rid of the denominator of a polynomial

I'm tutoring a high school precalculus student; our current topic is the roots of higher order polynomials. The problem we're solving is: Find a polynomial with the roots $\frac23$, -1, and $(3 + ...
0
votes
1answer
45 views

Polynomial (third degree)

A third degree polynomial $p(x)=0$ when $x=1$ and $x=3$. We also learn that $p(x) \geq 0 $ when $x \geq 1$ and $p(2) =2$. Determine $p(x)$. How should I proceed? I presume no calculus is needed.
1
vote
0answers
231 views

Finding irreducible polynomials and factorization

Need some explanation and checking if my thinking on the solution is correct for the assignment given below: (In these problems you may use without proof which polynomials of degree 2 and 3 are ...
1
vote
0answers
37 views

Given a cubic $f(x)$ with specified negative real roots $-a,-b,-c$, what happens when we search for solutions to $f(x)=d$?

Noting Roots of a Certain type of Cubic Equation, what if we have the following simpler form for real $d$: $$(x+a)(x+b)(x+c)=d\tag{1}$$ (With $a,b,c\in \mathbb R^+$.) Clearly, depending on $d$, the ...
1
vote
2answers
252 views

Help Factoring Quadrinomial

I know factoring questions are a dime a dozen but I can't seem to get this one. $-2x^3+2x^2+32x+40$ Factor to obtain the following equation: $-2(x-5)(x+2)^2$ Do I have to use division (I'd prefer ...
2
votes
3answers
521 views

Prove that $x^4-x-1$ is irreducible over $\mathbb{Q}$

Prove that $f(x)=x^4-x-1$ is irreducible in $\mathbb{Q}[x]$. All methods I know failed. I can only exclude that $f$ admits a factorization with a factor of degree 3, because in this case $f$ would ...
4
votes
2answers
98 views

Factorize $8x^3 + 12x^2 -2x -3$

How do I factorize this - $$8x^3 + 12x^2 -2x -3$$ I tried splitting the middle term but that didn't work , I tried factor theorem with various factors but even that didn't work. What can I do to ...
1
vote
1answer
45 views

Factorize $4a^2 - 9b^2 -2a - 3b$

I found this question in my textbook - $$4a^2 - 9b^2 -2a - 3b$$ I am in ninth grade and we have been taught how to factorize using identities, splitting the middle term and by using identities. I ...
0
votes
3answers
34 views

Factorize $(5a + \frac23)^2* (2a - \frac12)^2$

I found this question in my textbook - $$\left(5a + \dfrac23\right)^2\cdot \left(2a - \dfrac12\right)^2$$ I think it is already factorized but is there a way I can factorize it some more ? I tried ...
0
votes
2answers
55 views

Finding Factors Efficiently

Let $m$ and $n$ be positive integers. What is the most efficient way to choose factors that solve this equation. Notice that two factors of 2079 must sum to 36. What is a quick way of picking numbers? ...
4
votes
1answer
133 views

Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
4
votes
2answers
56 views

Prove irreducibility of a polynomial

Let $m$ be an integer, squarefree, $m\neq 1$. Prove that $x^3-m$ is irreducible in $\mathbb{Q}[X]$. My thoughts: since $m$ is squarefree, i have the prime factorization $m=p_1\cdots p_k$. Let $p$ be ...
0
votes
1answer
74 views

Polynomial Factorisation

Consider that we have a polynomial like $$x^3- (a + b +c ) x^2+abx-abc+s$$ Which is multiplication of $$(x-a)(x-b)(x-c)+s$$ Is it possible to reach value= $abc$ knowing the Coefficients and exponents ...