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

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74
votes
10answers
4k 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 ...
42
votes
16answers
50k 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 ...
28
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 ...
26
votes
2answers
2k 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 ...
23
votes
2answers
2k views

Is factoring polynomials as hard as factoring integers?

There seems to be a consensus that factorization of integers is hard (in some precise computational sense.) Is it known whether polynomial factorization is computationally easy or hard? One thing I ...
18
votes
4answers
783 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
422 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
568 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
9answers
2k views

Is $83^{27} +1 $ a prime number?

I'm having problems with exercises on proving whether or not a given number is prime. Is $83^{27} + 1$ prime?
15
votes
4answers
730 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
776 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
5answers
2k 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 ...
14
votes
2answers
162 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 ...
14
votes
1answer
543 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 ...
13
votes
10answers
4k views

Taking Calculus in a few days and I still don't know how to factorize quadratics

Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ...
12
votes
5answers
3k 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
8answers
606 views

How to factor quadratic $ax^2+bx+c$?

How do I shorten this? How do I have to think? $$ x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link to ...
11
votes
7answers
518 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 ...
10
votes
5answers
841 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 ...
10
votes
3answers
467 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 $\mathbb Z$ ...
10
votes
3answers
186 views

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

If $\begin{cases}a^4+b^4\in\mathbb Q\\ a^3+b^3\in\mathbb Q\\ a^2+b^2\in\mathbb Q\end{cases}$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. It is given that $a,b\in\mathbb R$. The proof of ...
9
votes
3answers
238 views

Intuitive understanding of the uniqueness of the Fundamental Theorem of Arithmetic.

Basically I am trying to understand why Fundamental Theorem of Arithmetic (FTA) exists, i.e why a natural number cannot be factored primely in two or more different ways. There are two proofs given ...
9
votes
5answers
365 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: ...
9
votes
1answer
159 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 ...
9
votes
0answers
236 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
8
votes
6answers
320 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
599 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
2answers
544 views

Is 641 the Smallest Factor of any Composite Fermat Number?

Consider the sequence $a_n = 2^{2^n}+1$ of so-called Fermat numbers. It's well known that $a_5$ isn't prime ($a_5 = 641 \cdot 6700417$, this is due to Euler). What I want to know about this sequence ...
8
votes
2answers
477 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
728 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
3k 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 ...
8
votes
0answers
103 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 ...
7
votes
4answers
310 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 ...
7
votes
7answers
615 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
605 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
114 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
323 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
118 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
1k 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
560 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 ...
7
votes
1answer
381 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
455 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
916 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
1k 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
273 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
266 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
5answers
81 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 ...
6
votes
4answers
1k 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
208 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... ...