0
votes
1answer
24 views

Factorizing $x^2-bxy+cy^2$

If $x^2-bx+c=(x+p)(x-q)$ ,then, factorise $x^2-bxy+cy^2$. My attempt - $(x+p)(x-q)=x^2+px-qx-pq$ $\implies p-q=b $ and $pq=c$ similarly-$p'-q'=by$ and $p'q'=cy^2$(assuming that on factorising ...
0
votes
1answer
44 views

Solve $x+\frac{2}{y}=3,y+\frac{2}{z}=3,z+\frac{2}{x}=3 $ in reals

find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right. $$ Things i have done: first i ...
3
votes
3answers
65 views

Prove $\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$ if $\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$

if $a,b,c$ are real numbers and $$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$$ Prove $$\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$$ things i have done: using the assumption i ...
1
vote
3answers
39 views

Solve $ x^2+y^2=4, z^2+t^2=9, xt+yz=6 $ in integers

find answers of this system of equations in integers$$ \left\{ \begin{array}{c} x^2+y^2=4 \\ z^2+t^2=9 \\ xt+yz=6 \end{array} \right. $$ things I have done: we can observe that ...
2
votes
2answers
31 views

Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ if $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$

if $a,b,c,d$ are positive real numbers and $$a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$$ Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ Things i have done: from assumption $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$ I ...
0
votes
1answer
29 views

Factoring $a^4(b-c)+b^4(c-a)+c^4(a-b)$

I was solving the question that wanted to factor $a^4(b-c)+b^4(c-a)+c^4(a-b)$. My idea was to factor a $(c-a)$ in first step.So $$b(a^4-c^4)+ac(c^3-a^3)+b^4(c-a)=a^4(b-c)+b^4(c-a)+c^4(a-b)$$ ...
6
votes
5answers
77 views

Prove $a^2+b^2+c^2=\frac{6}{5}$ if $a+b+c=0$ and $a^3+b^3+c^3=a^5+b^5+c^5$

if $a,b,c$ are real numbers that $a\neq0,b\neq0,c\neq0$ and $a+b+c=0$ and $$a^3+b^3+c^3=a^5+b^5+c^5$$ Prove that $a^2+b^2+c^2=\frac{6}{5}$. Things I have done: $a+b+c=0$ So ...
2
votes
2answers
46 views

For which $p$ and $q$ polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $F_p[x]$?

It easy to prove that polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $\mathbb{Q}[x]$ if $(q,6)=1$, since they don't have a common zero in $\mathbb{C}$, this can be seen geometrically. My question ...
1
vote
4answers
147 views

Factoring the following polynomials

Factorize the following polynomial: $t^3 -9t +8$ $t^6 -91t^2 +90$
2
votes
1answer
39 views

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
-1
votes
3answers
51 views

A problem on polynomial completely

$P(x)=x^3+mx^2+nx+14$ is divisible by $(x+2)$ but leaves a remainder of $-20$ when it is divided by $(x-2)$. Find the values of $m$ and $n$. Hence, factorise the polynomial completely. Now, I get ...
0
votes
2answers
60 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
0
votes
1answer
22 views

Proof the Existence and Uniqueness of Factorization Form of Polynomial with Complex Coefficient

If $p\in P(\Bbb{C})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form $$p(z)=c(z-\lambda_1)....(z-\lambda_m)$$ where ...
1
vote
1answer
19 views

Factoring homogeneous polynomials in two variables.

Consider a homogeneous polynomial $F(X,Y)\in\mathbb C[X,Y]$, why we can always write it as: $$F(X,Y)=\prod(a_iX+b_iY)^{r_i}\quad?$$ I can't find a proof of this fact. Many thanks in advance.
-1
votes
2answers
78 views

Factoring a problem. What is the other factor? [closed]

One of the factors of the polynomial $x^3-5x^2$ is $x+3$. What is the other factor
3
votes
2answers
71 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
81 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
72 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
43 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
53 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
60 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
26 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
46 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
30 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
60 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
52 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
72 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
17 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
26 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
22 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
136 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
22 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
92 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
40 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
522 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
78 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
164 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
42 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
65 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 $P(P(X))-X$
6
votes
3answers
244 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
197 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 ...