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

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78
votes
11answers
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 ...
0
votes
5answers
81 views

a more scientific way to factor

I used trial and error to factorise the following expression: ${9p^2 + 18p -16}$ I went through a number of different possibilities until I discovered the answer: ${(3p - 2)(3p + 8)}$ Is there an ...
0
votes
2answers
24 views

How to factor this expression completely?

$9x^2(4y^2-4y+1)-w^2z^2(4y^2-4y+1)$ I am ending up with $(2y-1)^2 (9x^2-w^2z^2)$. Can I go further? If so, I am unable to see it. Thanks
1
vote
1answer
16 views

Are there any “negative-factors” and “negative-multiples” as well, or are factors and multiples only positive numbers?

Basic question about Factors and Multiples: When we define factors as For an integer $x$, any integer which is completely divisible by $x$ is a factor of $x$. and multiples as For an ...
2
votes
3answers
47 views

Factorise the following expression?

So I need to factorise this expression but am a little stuck: $x^2+3(y+z)x+(y+2z)(2y+z)=?$ Anyone?
0
votes
0answers
16 views

Maximum degree of a polynomial [duplicate]

What is the maximum degree of a polynomial of the form $\sum_{i=0}^n a_i x^{n-i}$ with $a_i = \pm 1$ for $0 \leq i \leq n, 1 \leq n$, such that all the zeros are real? I have no idea where to start.
-2
votes
5answers
1k views

Proof of $a^n+b^n$ divisible by a+b when n is odd [on hold]

I read somewhere that $(a^n - b^n)$ It is always divisible by a-b. When n is even it is also divisible by a+b. When n is odd it is not divisible by a+b. and $(a^n + b^n)$ ...
-2
votes
0answers
30 views

Exercise about factorization

I've just started a new year at school, and I learned these formulas: $\sin x = \frac{e^{ix} - e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix} + e^{-ix}}{2}$ We used them in class to do some factorization ...
0
votes
1answer
25 views

Finding $|E|$, where $E$ is the Splitting Field of $x^8-1$ over a Field of $4$ Elements.

This is my attempt to find $\vert E \vert$, which is the order of the field $E$. If I am on the wrong track, please guide me to a technique that will work with more general fields and polynomials. We ...
2
votes
3answers
51 views

How to factor intricate polynomial $ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $

I would like to know how to factor the following polynomial. $$ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $$ What is the method i should use to factor it? If anyone could help.. Thanks in advance.
3
votes
3answers
77 views

Factorize Trigonometric Equation: $ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $

I have a problem with the following trigonometric equation: $$ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $$ It's from the book Engineering Mathematics 7th edition by Stroud. The book is giving the ...
0
votes
0answers
27 views

Why doesn't Horner's method work with the following cubic equation?

I'm trying to factor $$2x^3 - 4x^2 + 2x$$ I use the Horner's method │ 2 4 2 │ 0 --------------- │ 0 0 │ 0 --------------- 0│ 2 4 2 │ 0 and I obtain ...
5
votes
4answers
103 views

How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials?

Let $x,y$ be 2 coprime integers. I assume the following polynomial:$$f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$$ is not irreducible. So there must be at least 2 other polynomials of degree $\leq 4$ such that: ...
1
vote
1answer
395 views

Simplifying expressions - factoring or expanding?

The term "simplify" has always confused me. What does "simplifying" mean? More specifically, which is more "simple" - a fully factored expression or a fully expanded one?
1
vote
2answers
41 views

How to factor $ s^2LC + sRC + 2$

or $$ s^2+s\frac{R}{L}+\frac{2}{LC}=0 $$ Is there any way? I can't find out. Thanks in advance.
0
votes
2answers
64 views

Proof that every positive integer has at most one prime factor greater than it's square root?

I read the statement in the title somewhere but I can't find any proof. For a positive integer $n$, why can't there be 4 numbers $a, b, c, d$ ($b$ and $d$ are prime) for which $a < \sqrt{n} < ...
0
votes
1answer
41 views

Problem factorising a simple equation [closed]

I have to factorise the equation $x^2-y^2-z^2+2yz+x+y-z$. .How do I do it?Please help.
-1
votes
1answer
21 views

If A/B gives __ of 3, then B must be at least 4

I had an old note that I quickly wrote and I'm really unable to read the word after 'gives'. Can someone help me out? "If A/B gives __ of 3, then B must be at least 4"
0
votes
2answers
25 views

Minimum sum of factors of a natural number

Let's say I have a natural number $N$. $a$ and $b$ are two factors of $N$. How can I find $a$ and $b$ such that $a + b$ is minimum. Examples: $N = 12$, $a = 3$, $b = 4$ $N = 13$, $a = 1$, $b = 13$ ...
-1
votes
5answers
72 views
10
votes
4answers
106 views

How can you factor $x^4-6x^3+8x^2+2x-1?$

The original question is: Solve this equation for x: $$(x^2-3x+1)^2-3(x^2-3x+1)+1=x$$ I expanded and simplified it to get $$x^4-6x^3+8x^2+2x-1=0$$ Since neither -1 nor 1 are factors, it appears ...
4
votes
6answers
76 views

How can I factor the equation $a^3 - 5a^2 + 8a -4 = 0$?

Consider the equality $$ a^3 - 5a^2 + 8a -4 = 0 $$ What are the steps to factor this into the following? $$ (a − 1)(a − 2)^2 = 0 $$ Is there some technique to do this?
1
vote
2answers
66 views

Please the box method for factoring trinomial of the form $ax^2+bx=c$

I was given this method for factoring trinomials of the form $ax^2 + bx + c$ This is the method: Find numbers $p$ and $q$ such as $a\cdot c=p\cdot q$ and $b=p+q$ With $p$ and $q$ $(GCF(a,q)x ...
1
vote
1answer
54 views

Transforming the difference of cubes into a product

I want to know what happened at the denominator at line 2... Why did it it become like that, going from $x^3 - y^3$ to $(x-y)(x^2 +xy + y^2)$?
0
votes
1answer
23 views

Factoring question involving 4 integers.

For all non-negative integer values of $a$, $b$, $c$, $d$ given that $ac+bd+bc+ad=42$ and $c^2-d^2=12$, then determine all possible values of $a+b+c+d$. This was a question on one of my previous ...
0
votes
1answer
30 views

Analysis math irrational proof

How come $q = \frac{2p+2}{p+2}$ turns into $q^2 - 2 = \frac{2(p^2-2)}{(p+2)^2}$ I've tried factoring, square both sides, but I cant see it. this sample belongs to Rudin Analysis book
0
votes
2answers
45 views

Simplify: $\frac{a^4+3a^2b^2+b^4-2a^3b-2ab^3}{a^4+a^2b^2+b^4}$ [closed]

Simplify, $$\cfrac{a^4+3a^2b^2+b^4-2a^3b-2ab^3}{a^4+a^2b^2+b^4}$$ Maybe this is basic, but I'm stuck!
1
vote
1answer
28 views

factorisation of polynomial fraction

we have the fraction $\dfrac{x^2}{x^2-2x+1}$ (1) we can easily factorize the denomirator by using the polynomial's roots $\dfrac{x^2}{(x-1)^2}$ (2) now I saw somewhere that you can even simplify ...
2
votes
5answers
89 views

How to factor $4x^2 + 2x + 1$?

I want to know how to factor $4x^2 + 2x + 1$? I found the roots using quadratic equation and got $-1 + \sqrt{-3}$ and $-1 - \sqrt{-3}$, so I thought the factors would be $(x - (-1 + \sqrt{-3}))$ and ...
0
votes
2answers
409 views

what numbers multiply to 1 but add to negative 4

I have math hw on writing quadratic equations. You have to write them based on the parabola given in vertex form standard form and intercept/factored form. For the intercept form one step is to find a ...
0
votes
2answers
23 views

How can you factor this portion of the equation

This is a part of another person's question that was answered in the first question . I was confused on how they took $3x+4y+xy=2012$ and factored it out to $(y+3)(x+4)=2024$. If someone could explain ...
3
votes
0answers
21 views

What was Gauss' 2nd Factorization Method?

Reading Jean-Luc Chabert's A History of Algorithms, I learned that Gauss, prompted by the poor state-of-the-art, designed two distinct methods for fast integer factorization. Chabert's book discusses ...
1
vote
2answers
28 views

Factoring a polynomial with complex coefficients

Given $$3z^2+6z+3i=0$$ Find the complex roots and write in the form $a+bi$. I want to see how to factor it when there is an $i$ being multiplied by the constant.
0
votes
0answers
19 views

Finding An Equilibrium: Need to Factor a Complicated Polynomial

I am working on a game theory project/paper/exploration and an equilibrium condition reduces to finding a closed form solution for $y = f(n,k,a)$: $y^{n-1}-\binom{n-1}{k-1}y^{k-1}(1-y)^{n-k} = a$ ...
0
votes
0answers
20 views

Any way to factor, collect variable from this equation?

For a sum of quadratic solutions, is there any possible way to factor out the variable $P$ from the following real function? $QT$ is also a variable, and If it matters, $P > 0$ and all indexed ...
3
votes
3answers
83 views

Factor $k^{4}+4k^{3}+8k^{2}+8k+4=0$ over $\mathbb C$

Any idea how to factor the polynomial $k^{4}+4k^{3}+8k^{2}+8k+4$ over $\mathbb C$? Candidates for rational roots are $\pm1, \pm2, \pm4$ but none of them satisfies $k^{4}+4k^{3}+8k^{2}+8k+4=0$.
9
votes
0answers
87 views

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) = ...
1
vote
2answers
40 views

Questions from Dixon Factorization Paper

I read the first page of Asymptotically Fast Factorization of Integers, and have a few questions. Quotes from the paper are formatted as blockquotes (>). Following Legendre, we know that there ...
2
votes
5answers
34 views

Is it possible to factor a quadratic equation when $a$, $b$, and $c$ are all equal?

I have the equation $4x^2+4x+4$ to factor. I know that need to start with $$(2x \quad )(2x \quad )$$ to make $4^2$, but I can't seem to factor the rest of the way. What should I do?
1
vote
2answers
19 views

Factoring with fractional exponents

I don't know what it is, but this problem is giving me problems, although i have solved similar ones, this one is in all likely hood very simple, it's just : Solve for X $\sqrt{x} + (1 + \sqrt{x}) -2 ...
1
vote
2answers
24 views

Factoring out quotient of an expression.

I am following the MIT Open Courseware on Single Variable Calculus and in the first lesson when taking the derivative of a simple function I found myself confused because of my knowledge gap in ...
0
votes
0answers
19 views

Notation for separating out factors of a number

I have an integer (let's call it $n$), and I want to define it as the product of two values: one that's a pure power of two, and another that is odd. Obviously, these two values are unique for a ...
1
vote
1answer
47 views

polynomial factorization when exponent is not given

How can I factorise this equation, given i already know some of its factors which are: $(a-b)(b-c)(c-a).$ Equation is : $$a^nb^{n-1} + a^{n-1}c^n - a^nc^{n-1} - a^{n-1}b^n - b^{n-1}c^n + b^nc^{n-1}$$ ...
6
votes
5answers
10k views

How many positive integers are factors of a given number?

I've been trying to find / generate a formula for the following problem: Given a number, how many positive integers are factors of this number. In practice, you could simply build a table as such ...
1
vote
3answers
27 views

simplification of multiplying rational expressions

I cant' get the right answer to simplifying this expression: $$\frac{x^2+x-6}{x} \cdot \frac{x^2-3x}{x^2-9}$$ The answer I have is ${x - 2}.$ I can simplify it to $$\frac{x^2-6}{x} \cdot ...
0
votes
1answer
54 views

In Maple, how can I partially factor a lengthy symbolic expression (23 terms in 6 variables)?

I need to show that the following expression, $$a^3b-a^3c+a^3z+a^3x+a^3y-a^2bx+a^2by+a^2cx-a^2cy-a^2zx+a^2zy-a^2x^2+a^2y^2-abcz-abcx-aczx-acx^2+b^2c^2+2bc^2x+c^2x^2-b^2c-2bcx-cx^2,$$ is positive ...
0
votes
1answer
81 views

Factorizing a cubic polynomial

This is the result of determinant evaluation: $$p(x) = (x-3)((x-1)(x-2)-1)+1$$ How can I factor this polynomial?
0
votes
4answers
89 views

Factorizing Faster

As one of the steps in a calculation, I had to factorize the following quadratic polynomial: $$12x^2 - 11x - 15$$ Using guess and check, and after roughly 10 minutes, I came to the following ...
2
votes
5answers
88 views

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals (A) $\frac{1-q^{128}}{1-q}$ (B) $\frac{1-q^{64}}{1-q}$ (C) $\frac{1-q^{2^{1+2+\dots +6}}}{1-q}$ ...
1
vote
2answers
29 views

Methods for verifying correct factorisation of polynomials

In an attempt to factor using a GCF, Mia wrote $8x^2 + 4x = 4x(2x – 0)$, which is not correct. a. Explain how Mia could check her work. b. What error did Mia make? She didn't factor using the GFC ...