For questions about finding factors of e.g. integers or polynomials

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68
votes
10answers
3k views

Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and ...
39
votes
14answers
33k views

What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We are factoring polynomials and we both had the same thought at the same time: when are we going to use this? I feel ...
31
votes
4answers
1k views

Could G. H. Hardy make a product of two primes so big he couldn't find out which?

This question of course began as a slightly irreverent play on the riddle, "Can God make a stone so big He can't lift it?" Then I wondered about the answer. Is it possible to exhibit a number that is ...
25
votes
2answers
1k views

Factorize $(x+1)(x+2)(x+3)(x+6)- 3x^2$

I'm preparing for an exam and was solving a few sample questions when I got this question - Factorize : $$(x+1)(x+2)(x+3)(x+6)- 3x^2$$ I don't really know where to start, but I expanded everything to ...
25
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
17
votes
4answers
725 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
16
votes
1answer
373 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 $\mathbb{Q}$? ...
16
votes
4answers
528 views

Are polynomials of the form : $ f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a$ irreducible over $\mathbb{Z} $?

Is it true that polynomials of the form : $ f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a$ where $\gcd(n+1,k+1)=1$ , $ a\in \mathbb{Z^{+}}$ , $a$ is odd number , $a>1$, and ...
15
votes
4answers
703 views

Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$

Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
15
votes
4answers
681 views

Factor $x^4 - 11x^2y^2 + y^4$

This is an exercise from Schaum's Outline of Precalculus. It doesn't give a worked solution, just the answer. The question is: Factor $x^4 - 11x^2y^2 + y^4$ The answer is: $(x^2 - 3xy -y^2)(x^2 + ...
14
votes
2answers
158 views

Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$

How do I simplify: $$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$ Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
12
votes
5answers
2k 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 + ...
11
votes
5answers
1k views

Factoring a hard polynomial

This might seem like a basic question but I want a systematic way to factor the following polynomial: $$n^4+6n^3+11n^2+6n+1.$$ I know the answer but I am having a difficult time factoring this ...
10
votes
7answers
501 views

Can someone show me why this factorization is true?

$$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \dots + xy^{n-2} + y^{n-1})$$ Can someone perhaps even use long division to show me how this factorization works? I honestly don't see anyway to "memorize ...
9
votes
5answers
270 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: ...
8
votes
6answers
314 views

Find $x$ and $y$ in $2^{x-y} + 1 = 2^x,$ where $x,y$ are integers

I have no idea what to do now. Is there any way to find the integers $x$ and $y$ by factoring? Thank you.
8
votes
7answers
583 views

Solve $\sqrt{x+4}-\sqrt{x+1}=1$ for $x$

Can someone give me some hints on how to start solving $\sqrt{x+4}-\sqrt{x+1}=1$ for x? Like I tried to factor it expand it, or even multiplying both sides by its conjugate but nothing comes up ...
8
votes
5answers
541 views

Factor $(a^2+2a)^2-2(a^2+2a)-3$ completely

I have this question that asks to factor this expression completely: $$(a^2+2a)^2-2(a^2+2a)-3$$ My working out: $$a^4+4a^3+4a^2-2a^2-4a-3$$ $$=a^4+4a^3+2a^2-4a-3$$ $$=a^2(a^2+4a-2)-4a-3$$ I am ...
8
votes
2answers
349 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 = ...
8
votes
4answers
646 views

How to determine in polynomial time if a number is a product of two consecutive primes?

How to determine in polynomial time if a number is a product of two consecutive primes? All I can figure out is that if Cramér's conjecture is true, then we can use the AKS primality test to find ...
8
votes
1answer
141 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 ...
8
votes
1answer
2k views

Pollard-Strassen Algorithm

I'm aware that the Pollard-Strassen algorithm can be used to find all prime factors of $n$ not exceeding $B$ in $O\big(n^{\epsilon} B^{1/2}\big)$ time. This is really useful because I need to find all ...
7
votes
7answers
363 views

If $a^3 + b^3 +3ab = 1$, find $a+b$

Given that the real numbers $a,b$ satisfy $a^3 + b^3 +3ab = 1$, find $a+b$. I tried to factorize it but unable to do it.
7
votes
3answers
110 views

How to see that the shift $x \mapsto (x-c)$ is an automorphism of $R[x]$?

In the process of studying irreducibility of polynomials, I encountered the criterion that $p(x)$ is irreducible if and only if $p(x-c)$ is irreducible. When trying to determine what properties of the ...
7
votes
3answers
481 views

How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.

I would please like some help with solving the following equation: $$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$ All I know about the equation is that there is a root with the real part of $1$. My approach ...
7
votes
3answers
141 views

Is $t^4+7$ reducible over $\mathbb{Z}_{17}$?

Is $f=t^4+7$ reducible over $\mathbb{Z}_{17}$? Attempt: I checked that $f$ has not roots in $\mathbb{Z}_{17}$, so the only possible factorization is with quadratic factors. Assuming ...
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 ...
7
votes
1answer
801 views

Even numbers have more factors than odd numbers…

This was an exercise to show that, in a sense, the even numbers have more prime factors than the odds, but--if it's right-- I still have a question. As an heuristic calculation, we could take a large ...
7
votes
1answer
333 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 ...
7
votes
1answer
357 views

factorise, $x^3-13x^2+32x+20$

factorise, $x^3-13x^2+32x+20$ Let, $f(x)=x^3-13x^2+32x+20$ $f(x)=x(x^2-13x+30)+2x+20$ $f(x)=x(x-3)(x-10)+2x+20$ $f(-1)\lt 0$, $f(0)\gt 0$, which shows there is a root between $x=-1$ and $x=0$ ...
6
votes
6answers
835 views

How to factor $2x^2-x-3$?

I know its: $$(x+1)(2x-3)$$ But how do you come to that conclusion?
6
votes
5answers
945 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.
6
votes
1answer
260 views

Find the value of $x^3-x^{-3}$ given that $x^2+x^{-2} = 83$

If $x>1$ and $x^2+\dfrac {1}{x^2}=83$, find the value of the expression$$x^3-\dfrac {1}{x^3}$$ a) $764$ b) $750$ c) $756$ d) $760$ In this question from given I tried to ...
6
votes
4answers
245 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!
6
votes
4answers
989 views

Find all real solutions to $8x^3+27=0$

Find all real solutions to $8x^3+27=0$ $(a-b)^3=a^3-b^3=(a-b)(a^2+ab+b^2)$ $$(2x)^3-(-3)^3$$ $$(2x-(-3))\cdot ((2x)^2+(2x(-3))+(-3)^2)$$ $$(2x+3)(4x^2-6x+9)$$ Now, to find solutions you must set ...
6
votes
2answers
156 views

Simple factoring in proof by induction

How would this: $$\frac{((n+1)+1)(2(n+1) + 1)(2(n+1) + 3)}{3}$$ Factor to this: $$(2(n+1)+1)^2$$ This is a part of an induction proof, which I would post an image if my reputation was higher... ...
6
votes
3answers
256 views

Finding the number of factors of product of numbers

If $a,b,c,d\in\mathbb{N}$. Each of which has exactly five factors, can we determine the number of factors of the product of $a,b,c,d$? Edit This is the solution given the in the back of the book I ...
6
votes
3answers
87 views

How do I factor this?

How do I factor $p^2+8pq+16q^2-9r^2$? I know how to group the first two terms, but I dont know what to do with the other half. Can someone help me with this problem?
6
votes
1answer
705 views

Simply Explain the General Number Field Sieve

As a beginner to the world of integer factorization, my idea of factoring an integer is to generate a large list of prime numbers below this number and to repeatedly try to divide the integer by these ...
6
votes
3answers
8k views

Find the largest prime factor

I just "solved" the third Project Euler problem: The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ? With this on Mathematica: ...
6
votes
1answer
457 views

Smallest number with a given number of factors

From my rather rudimentary explorations of this fascinating problem, I believe it to be a layered and rewarding subject for investigation. My question, essentially, is: How do you find the smallest ...
6
votes
2answers
482 views

Is the factorization problem harder than RSA factorization ($n = pq$)?

Let $n \in \mathbb{N}$ be a composite number, and $n = pq$ where $p,q$ are distinct primes. Let $F : \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$ (*) be an algorithm which takes as an input $x ...
6
votes
0answers
79 views

Is there an easy way to factor polynomials with two variables?

On a recent precal test, I saw a question involving the following expression: $$(x+1)^2-y^2$$ Which factored out into: $$(x+y+1)(x-y+1)$$ This wasn't very hard, considering that it was already ...
6
votes
1answer
224 views

Factoring some integer in the given interval

Let N be a positive integer. Is there an efficient (i.e. probabilistic polynomial time) algorithm which, on input a sufficiently large N, outputs the full factorization of some integer in the interval ...
5
votes
3answers
212 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 ...
5
votes
6answers
7k views

How do you factor $x^3-3x^2+3x-1$?

$$x^3-3x^2+3x-1?$$ I know this may seem trivial, but I, for the life of me, I cannot figure out how to factor this polynomial, I know that the root is $$(x-1)^3=0$$ because of wolframalpha, but I ...
5
votes
6answers
259 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. ...
5
votes
5answers
340 views

How to show $x^4 - 1296 = (x^3-6x^2+36x-216)(x+6)$

How to get this result: $x^4-1296 = (x^3-6x^2+36x-216)(x+6)$? It is part of a question about finding limits at mooculus.
5
votes
2answers
311 views

Factoring $X^{16}+X$ over $\mathbb{F}_2$

I just asked wolframalpha to factor $X^{16}+X$ over $\mathbb{F}_2$. The normal factorization is $$ X(X+1)(X^2-X+1)(X^4-X^3+X^2-X+1)(X^8+X^7-X^5-X^4-X^3+X+1) $$ and over $GF(2)$ it is $$ ...
5
votes
6answers
226 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 ...