Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [factoring]

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

If an element $n$ of the integers or more generally of a ring, e.g. a ring of polynomials, can be written as $n=ab$ with $a,b$ in the same ring, then $a,b$ are factors or divisors of $n$. If one of the factors is a unit (a divisor of 1), the factorization is called trivial. Finding a non-trivial factorization (if one exists) can be a daunting task and is the aim of many algorithms. A good factorization method might even render some strong encrpytion methods obsolete.

Having such a factorization can be very helpful, e.g. to simplify tasks: Finding the roots of $x^5-3x^2-5x^2+15$ may be difficult without knowing the factorization $(x^2-3)(x^3-5)$.

The (factoring) tag is suitable for questions about finding factors of a specific number, polynomial etc. or about aspects of general factorization methods or the structure of the set of factors.

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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 polynomial properly. (It should be $(n^2 + 3n + 1)^2$).…
InsigMath
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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 written as the difference of two squares. I then…
PhiNotPi
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Factoring $x^n + y^n$ over the integers

Here's what i know (or think i know) about the factoring. For integer $n> 1 $ 1) If $n$ is a positive power of $2$ then it is irreducible. 2) If $n$ is an odd prime then $$x^n + y^n = (x + y)(x^{n-1} - x^{n-2}y + \cdots - xy^{n-2} + y^{n-1} ) $$ 3)…
neofoxmulder
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Factorize: $a^3(b+c)+b^3(a+c)+c^3(a+b)$.

Factorize: $a^3(b+c)+b^3(a+c)+c^3(a+b)$. I found this question on a high school textbook but it seems impossible to be further factorized. The best I can get is: $a^3(b+c)+b^3(a+c)+c^3(a+b) = (a+b+c)(a^3+b^3+c^3)-a^4-b^4-c^4$. Since this question is…
Henry Ng
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Determine all factors of zero (divisors of $0$)

How many integer factors of $0$ are there, and what are they? I'm just curious, but what counts as a factor of $0$? My guess is that there are an infinite number of factors of $0$, but is there a proof?
Jason Chen
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Factorise $f(x) = x^3+4x^2 + 3x$

Not sure if this belongs here, but I'm slowly trudging through my studies for Math 1 and wondered if y'all could give feedback and/or corrections on the following factorisation question: $$ \text{Factorise}: f(x) = x^3+4x^2+3x $$ Firstly, the GCD of…
Dan
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Need help solving a bi-quadratic polynomial....

The polynomial to be factorised as a product of two factors is- $$x^4+3x^2+6x+10$$. I checked the solution in wolfram alpha to be- $$(x^2-2x+5)(x^2+2x+2)$$. I tried to factorise it by expressing it as a sum of two squares $$(x^2+1)^2+(x+3)^2$$. But…
Soham
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Factorize: $a^2(b − c)^3 + b^2(c − a)^3 + c^2(a − b)^3$

I want to factorize $a^2(b − c)^3 + b^2(c − a)^3 + c^2(a − b)^3$ . By inspection , I can see that substituting $b$ for $a$ yields $0$ thus $(a-b)$ is a factor . Similarly $(c-a)$ and $(b-c)$ are factors . But I can't figure out other factors. Yes ,…
A Googler
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How to factor $y = x^5 + 20x^2 + 5$?

How would I factor to solve for x? $x^5 + 20x^2 + 5=0 $? Do I use synthetic division? Is there a faster/easier way? Do I have to keep plugging in numbers to see if they equal to zero? Thanks! I'm not asking for full solutions if you don't want to…
Jessica
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Factorise: $2a^4 + a^2b^2 + ab^3 + b^4$

Factorize : $$2a^4 + a^2b^2 + ab^3 + b^4$$ Here is what I did: $$a^4+b^4+2a^2b^2+a^4-a^2b^2+ab^3+b^4$$ $$(a^2+b^2)^2+a^2(a^2-b^2)+b^3(a+b)$$ $$(a^2+b^2)^2+a^2(a+b)(a-b)+b^3(a+b)$$ $$(a^2+b^2)^2+(a+b)((a^2(a-b))…
A Googler
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Factorize $a^2-ab-bc\pm c^2$

I got this question in a test but it did not specify the variable with respect to which I was supposed to factorize $$a^2-ab-bc\pm c^2$$ where it could be just $a(a-b)-c(b\pm c)$ but no common factor over all terms. I feel I may be missing…
hhh
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How to factor $x^4 +3x -2$?

I have figured out there is two roots between $0$ and $1 ,-1$ and $-2$ for $x^4 +3x -2 = 0$. Therefore there should be two factors $(x + a)$ and $(y - b)$ where $a,b \in R^+$. But how to find these $a$ and $b$? When they found I can find the next…
lakshman
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What does it mean to factor over the real numbers?

I am confused on the topic of factoring over real numbers. What is the difference between normally factoring and factoring over real numbers? If anyone could explain, that would be appreciated! Thanks ahead of time!
James
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Factor $(x+y)^4+x^4+y^4$

Title says it all, I just want to know how to factor $(x+y)^4+x^4+y^4$. I only know that it's possible to factor, but got no idea how to do it. If it were a single-variable polynomial I could try to find rational roots or something, but I'm lost…
Random Dude
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Rational equation

my question İf $ x+\frac{1}{x^2}=3$ then find $( x^2 -\frac{1}{x})^2 $ . I tried factoring, taking squares of both sides and some other things that did not work. what should i do?
matbaz
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