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

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2
votes
4answers
69 views

Why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$

I would like to understand why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$ I am particularly concerned with the term, $-4$.
0
votes
1answer
23 views

Factorizing Given Problem

I have searched through various site's and forums but couldn't find the answer to my problem, $$z^2-\frac{1}{2}z-\frac{1}{4}=0$$ How will you factorize this As I can't find $2$ numbers that give me ...
4
votes
2answers
73 views

Factoring in $\mathbb{Z}[\sqrt{2}]$

How would one factor a number, say $9+4\sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$? This is what I've attemped to do: $$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2}) $$ $$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$ ...
0
votes
1answer
24 views

Integer factorization complexity

Why isn't the problem of factoring an integer known to be in $P$? Isn't the naive algorithm of trying to divide a number by all the numbers up to its squre root polynomial?
2
votes
1answer
41 views

Possible values of $\gcd(a+b, a\times b)$

Main Question: Let $N \in \mathbb{N}$. What are the possible values of $\gcd(a+b, a\times b)$ given that $\gcd(a,b) = N$? Fact 0. If $\gcd(a,b) = N$, then $N \leq \gcd(a+b, a\times b) \leq ...
0
votes
1answer
22 views

Word Problem involving Factorization of a Polynomial

I know how to factor this polynomial, but I'm not sure how the factors of the function relate to the possible dimensions of the aquarium. I'm not sure where to start, to be honest. Any ideas would be ...
0
votes
2answers
27 views

How to extract factor when expression is with a power

$$f(x) = x^2(2x-3)^3$$ I tried to extract the 2 from the parenthesis. $$f(x) = 2x^2(x-\frac{3}{2})^3$$ But the graphic from this function is different. What should I consider when doing this kind ...
1
vote
1answer
39 views

Prime factorization difficulty

From Wikipedia: Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime ...
0
votes
0answers
25 views

Show a curve has no factor of degree 1 or 2

I have to show that $ h(x,y)=y^{2}(x^{2}+x+1)-x^{2} $ has no factors of degree 1 or 2. I know that h contains infinitely many points and is singular at the points (1,0,0), (0,1,0) and (0,0,1). I am ...
-4
votes
1answer
83 views

limit of function at $x \rightarrow 2$

ok, so this is a very basic question, i'm trying to find the limit of the following function at $x \rightarrow 2$: $|x^2 + 3x + 2| / (x^2 - 4)$ what i had previously done was simply plug in 2 for ...
1
vote
2answers
43 views

limit of square root function at $x \to 6$

i'm trying to find the limit of the following function at x -> 6: $$\frac{x^2-36}{\sqrt{x^2-12x+36}}$$ i've simplified it so that it becomes $\dfrac{(x+6)(x-6)}{\sqrt{(x-6)^2}}$, which simplifies to ...
-4
votes
3answers
93 views

Help With Factoring a Seventh Degree Polynomial [closed]

Factor the seventh degree polynomial $x^7 + x^2 + 1$
1
vote
1answer
17 views

Factoring completely using complex cube of unity

How can you completely factor $a^2 + ab + b^2$ and $a^2 - ab + b^2$ completely using $\omega$, the complex root of unity? Is there some general rule for such complex factorisations? Any help would be ...
1
vote
4answers
84 views

Show that $(k!)^n$ divides $(kn)!$

Show that $(k!)^n$ divides $(kn)!$ I've tried it but without success. Any help would be great.
1
vote
1answer
36 views

Multiplying two fractions with complex numbers

I'm doing $$ \frac{6-7i}{1+i}\cdot\frac{1+i}{1+i}, $$ and I'm getting the correct value for the numerator (namely, $-1-13i$), but based on the problem answer, I need for the denominator to become $2$. ...
1
vote
2answers
39 views

Simplifying an inequality: $4x(x-2) \lt 2(2x-1)(x-3)$

I have: $$4x(x-2) \lt 2(2x-1)(x-3)$$ For the last part, do I multiply both things in $()$ by two then solve them like I normally would? If I solve them and then multiply will it work the same? Is that ...
2
votes
0answers
77 views

Humankind knows the prime factorization of the first how many consecutive integers?

I am only looking for an approximation. I'm guessing the answer must be somewhere between $10^{20}$ and $10^{50}$. . Edit: Okay so my first initial estimation was pretty poor... I should have ...
0
votes
3answers
59 views

Factoring Real and Complex polynomials.

Factor: a) $x^2 + 1 \in \mathbb{R}[x]$ b) $z^3 - i \in \mathbb{C}[x]$ Well I solved for $x^2$ and got $-i$ and $i$, but they aren't from Real. And I couldn't solve for Complex (part b).
0
votes
2answers
22 views

Basic complex factorisation

Let's say I want to find all the roots of $f(z)=z^8-256$. Factorising it, I find $f(z)=(z-2)(z+2)(z^2+4)(z^4+16)$. $z =\pm2,\,\pm2i$ is only 4 roots. Shouldn't there be another 4?
0
votes
4answers
46 views

Solving for x by completing the square in a problem where the solution doesn't seem to have symmetrical answers

So I've been given this problem: $-14x^2 + 45x + 14 = 0$ And I've tried it a number of times but can't seem to solve it. The answer is supposed to be found by completing the square, and the solution ...
0
votes
1answer
28 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 ac=pq and b=p+q With p and q (GCF(a,q)x + GCF(c,p))(GCF(a,p)x+GCF(c,q))= ...
2
votes
2answers
50 views

Factorize the given equation.

Factorize $$f(t) = t^3 - 11t^2 - 39t - 45$$ Assuming the above polynomial has a rational root, I tested the above equation using $+1$, $-1$, $+2$, $-2$. These did not work out. Then I tried $t =3$. ...
0
votes
2answers
20 views

Question about factoring/condensing equation rules

I have the equation $x^2 - 6x = 72$ and then $x^2 - 6x - 72 = 0$ that's supposed to turn into $(x-12)(x+6)$. 72/6 = 12. So could just do that with any equation that? Divide the end thing with the ...
0
votes
1answer
33 views

Calculate sum of all factors of expression

Expression: $$ \left(\frac{2x}2\right)^2 \left(\frac{3y}3\right)^3$$ Sum of all factors of above expression is $$2\cdot \left(\dfrac{2x}2\right) + 3\cdot\left(\dfrac {3y}3\right)$$ How ? Can ...
0
votes
1answer
21 views

Deflating (factoring) a 6th degree polynomial

What is the procedure to factor a 6th degree polynomial of a complex variable? $$P(z)=1+x^2+x^3+x^4+x^5+x^6$$ I do have the correct answer but no idea how to get to it. The answer is: ...
0
votes
2answers
44 views

Direct Proof even and odd

In trying to show that $n$ is even, is my final solution correct? First: If $n$ is even then $n^3+n$ is even. Since $n$ is even, then: $$n=2\cdot s$$ $$n^3+n = (2\cdot s)\cdot (2\cdot s)\cdot (2\cdot ...
0
votes
3answers
47 views

tough factorisation problem

How would you factorise this equation given that $x=7$ is a root of this equation $$x^3 - 67x + 126 = 0.$$ Any help would be thoroughly appreciated.
1
vote
1answer
17 views

Polynomial Factoring over a finite field

Ok, so I'm trying to figure out how to factor polynomials over a finite field. My polynomial is x^5 + x^2 + x + 1 and I have to factor over GF(2) I know the answer is (x+1)^2 * (x^3 + x + 1), because ...
0
votes
0answers
30 views

Solving equation in maxima not placing variable on one side

I'm trying to solve an equation but the variable ($\varphi$ PHI) will not factor out to one side. Is there any other way to do this? I'm using maxima version 5.32.1 Here's the equation in latex as ...
6
votes
3answers
117 views

Polynomial factorisation - absolute value of coefficients

This question takes the factorisation of a polynomial $p(x)=q(x)r(x)$, where $p$ (and for my purpose here $q$ and $r$) have integer coefficients and asks if the maximum absolute value of the ...
1
vote
2answers
29 views

How to find an alternate form of this polynomial (factorize?)

I am trying to find the limit of the function $$\lim_{t \to 1} {{t^3-2t+1}\over{t^3+t^2-2}}$$ And it obviously evaluates to ${0\over0}$ so at first glance it is indetermined. But I have these two ...
1
vote
3answers
193 views

Factoring the Polynomial $x^4-2x^2+1$

Okay, I am practicing factoring for an upcoming assignment and I know that this is basic algebra, but I forgot how to attack this polynomial. Every method that I have used so far from simply guessing ...
0
votes
0answers
22 views

Integer Factorization

if $x \not \equiv \pm y$ (mod $n$) and $x^2 \equiv y^2$ (mod $n$), then $\gcd(x \pm y, n)$ are factors of $n$. Proof: $x^2 \equiv y^2$ (mod $n$) $\Rightarrow n$ is a factor of $(x-y)(x+y)$. Note ...
1
vote
0answers
33 views

Binary Polynomial Factoring

I just need confirmation that I've done my math right. If $a(x) = x^4 + x^3 + x + 1$ and $b(x) = x^2 + x + 1$ are binary polynomials, find binary polynomials s(x) and r(x) such that $x^4 + x^3 + x + ...
2
votes
4answers
75 views

Solving $y^2 - yx - y + x = 0$ for $y$?

I solved this equation for $y$ by inspection and confirmed it with Wolfram Alpha - $y^2 - yx - y + x = 0$ I got the values $y = 1$ and $y = x$ However I was wondering is there a formal method for ...
0
votes
2answers
24 views

Solving a polynomial equation by factoring

The polynomial f(x) is defined by $$f(x) = 12x^3+25x^2 -4x -12$$ (i) Show that f(-2) = 0 and factorise f(x) completely. Which i did and got $(x+2)(3x-2)(4x+3)$ (ii) Given that $$12 * 27^y + 25 * ...
0
votes
1answer
17 views

Factoring binary polynomials

I need to factor two binary polynomials and present each as a product of powers of irreducible polynomials. a) x⁴ + 1 I have figured it out this far: x⁴ = (x²)² and 1 = 1² So I have something in ...
0
votes
1answer
15 views

how to find the interval at which a derivative function is increasing

Alright, so here's the deal. I need to find the interval of this derivative function: f(x)= −5x2+12x−7 So far, I've gotten that the derivative is this: ...
-1
votes
1answer
32 views

Factoring/Expansion explantion

Sorry if I call something by the wrong name since I didnt learn math in english. ok so for example this: (a+b)(a-b) if you break it down to the second "()" you will end up with this: a+-b could ...
4
votes
3answers
66 views

How to factor $x + 1 - 2 \sqrt x$?

My teacher said the answer is $(\sqrt x -1)^2$, but I want to know how he figured it out. I know it's a trick I learned years ago, but I can't remember how to do this.
0
votes
0answers
18 views

Factoring big numbers into primes

I can't find a good tutorial anywhere on how to factor big numbers into primes, so I was wondering if someone could explain the process. I need to do this for my cryptology class.
1
vote
1answer
19 views

Evaluating cubic roots of a quadratic

If $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^2 + 4x -5 = 0$, evaluate $\alpha^3 + \beta^3$.. I know that $$\alpha + \beta = \frac{-b}{a}$$ and $$\alpha \beta = ...
-1
votes
2answers
40 views

Fully factorise $x^3-x^2-14x+24$ into linear factors

$$f(x)=x^3-x^2-14x+24$$ I've tried grouping the terms, but it just doesn't work out for me. Any help is appreciated.
1
vote
2answers
33 views

Factoring somewhat complex polynomial

Can this be factored? $$m^2(a-1) + 2m(a-1) -1$$ I have not been able to find any roots that will work when multiplied out. Does anyone see any options?
0
votes
1answer
28 views

yacas factorize polynoms

I want factorize polynoms with yacas but I can do it only with univarial. E.g. I want $x^2-y^2$ factorize to $(x-y)(x+y)$. How can I do it? Or if anybody has any suggestion to another simple, free ...
1
vote
2answers
28 views

Factor the Quadratic

-16t^2+32t+20=0. How are you supposed to find -5 and positive 1 to put in the parenthesis? -4(2t-5)(2t+1)?
2
votes
3answers
74 views

Finding all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors.

Find all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors. I've tried checking the first 6-7 $n$'s on wolframalpha, but I don't see any patterns for even nor odd $n$'s. At first ...
0
votes
2answers
26 views

Completely factor a polynomial using the rational root theorem and synthetic division

I am currently seriously confused. My problem, as stated above, is about completely factoring a polynomial. My question is, once you get your possible factors, how do you then simplify it down? Ill ...
0
votes
1answer
25 views

Regarding +/- fractions: what are some mental tests you can apply to uncommon fraction denominators?

When adding and subtracting fractions: what if there is no uncommon factor (for example 4=2,2 and 6=2,3). Does that always mean to use the LCM? What if the LCM is too big or time consuming to ...
1
vote
3answers
66 views

Factorize $6x^2 -5x -14 = 0$

I'm throwing a bit of a blank on the best way to factor this : $$6x^2 -5x -14 = 0$$ I know that I could multiply $6$ by $14$ and then find a pair of factors that add to $-5$ (b), but this feels a ...