3
votes
2answers
60 views

prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
2
votes
2answers
80 views

Is there a reason for some polynomial quotients to have a remainder equals to zero?

I was helping some highschool students with factorization exercises. They had alternatives to choose the correct factor. Then one of them said to me: We use a calculator and evaluate some prime ...
0
votes
3answers
65 views

prove $(a+b+c)^n=a^n+b^n+c^n$ if $(a+b+c)^3=a^3+b^3+c^3$

if $(a+b+c)^3=a^3+b^3+c^3$ and n is odd number,prove that: $$(a+b+c)^n=a^n+b^n+c^n$$ hint of the question was: factor this expression $f(a,b,c)=(a+b+c)^3-(a^3+b^3+c^3)$ after factorization ...
1
vote
3answers
31 views

Factor Cyclic Polynomial

Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
3
votes
2answers
48 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
0
votes
2answers
56 views

give a complete factored form of the polynomial $-6a^5+48a^4+12a$

Give a complete factored form of the polynomial $-6a^5+48a^4+12a$ I have tried solving this equation and I just cant figure it out. Help me, and give me the answer.
0
votes
3answers
12 views

Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
0
votes
1answer
41 views

Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
0
votes
1answer
28 views

remainder is not zero using long division method

Find all zeros of $f(x)=128x^3-48x^2+1$ given that one linear factor occurs twice. let $f(x) $ be equaal to 0 $128x^3-48x^2+1=0,$ $16x^2(8x-3)+1=0,$ trying $x=1/4$ $16/16(2-3)+1=0,$ ...
0
votes
1answer
54 views

Factoring $x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2$

The subject line pretty much says it all. In my geometry class today, the following equation came up: $$x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2 = 0$$ Specifically, it was in the ...
2
votes
1answer
84 views

What do we know about $\displaystyle \frac{f}{\gcd(f,f')}$ if $f\in\mathbb{F}_{p^d}[X]$?

Let $\mathbb{K}=\mathbb{F}_{p^d}$ and $f\in\mathbb{K}[X]$ be a non-constant polynomial with the factorization $$f=\prod_{i=1}^nf_i^{k_i}$$ where $f_i\in\mathbb{K}[X]$ is irreducible and ...
1
vote
1answer
27 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
-1
votes
6answers
49 views

Polynomial factors involving inequalities

How to factorise the polynomial $p(x) = x^4-2x^3 + 2x - 1$. Hence, solve the inequality $p(x) \gt 0$ ?
3
votes
2answers
71 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
2
votes
1answer
14 views

Polynomial factorisation on integers modulo n

Is there a known (efficient) algorithm to compute the list of factors of a polynomial modulo $n$ (for any integer $n$)? For example in $\mathbb Z_8$, $X^2+2X$ has a list of 4 factors (multiplicity 1 ...
0
votes
1answer
25 views

Factoring Polynomial Questions

How do you decide whether to use synthetic division or the factor theorem to help you factor a polynomial? Please help me answer.
0
votes
1answer
19 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
6
votes
1answer
132 views

Factorise $x^4 + 3x^2+ 6x+ 10$

I need to factorise $x^4 + 3x^2 + 6x + 10$ completely over $\mathbb{Q}$. I am not sure how to do this. I can't find any roots of this equation in $\mathbb{Z}$.
1
vote
0answers
20 views

Factorisation algorithm for polynomials in several variables over $\mathbb{Z}$.

What algorithm is used by a CAS to decide how to factor a polynomial in several variables over $\mathbb Z$?
1
vote
2answers
79 views

Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck. The problem is that I can not see the telescoping pattern anywhere for following problem. Prove that $$a^n - b^n ...
0
votes
1answer
66 views

Given a polynomial of degree 5, get minimum and maximum without using derivatives

Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial ...
0
votes
2answers
35 views

Factorisation of a polynomial [closed]

I have a polynomial $$t^4-4\lambda t^2-4t^2 $$ I need to give a real value to $\lambda$ such that i get 4 real roots.
1
vote
1answer
37 views

How to give irreducible factorization in $\mathbb{Z}_5$?

I have this polynomial: $$f=x^5+2x^3+4x^2+x+4$$ How can i find the irreducible factorization(in $\mathbb{Z}_5$)?I can find the roots easily but thats not enough.
14
votes
1answer
512 views

Is factoring polynomials easier than factoring integers? [duplicate]

I was reading the book Algebra: Chapter 0 , by Paolo Aluffi, and came across the following assertion, in page 290, Exercise 5.9: It is in fact much harder to factor integers than integers ...
0
votes
2answers
24 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
75 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
23 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
155 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
37 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
64 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
233 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
141 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
57 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
56 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
52 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
131 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
65 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
68 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
52 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
118 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
164 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
35 views
1
vote
2answers
75 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
70 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
61 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
60 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
89 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
33 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?