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

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2
votes
4answers
93 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
29 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
82 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
33 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
54 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
2answers
31 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
51 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 ...
-4
votes
1answer
90 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 ...
3
votes
2answers
61 views

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

I'm trying to find the limit of the following function at $x \to 6$: $$\frac{x^2-36}{\sqrt{x^2-12x+36}}$$ i've simplified it so that it becomes $\dfrac{(x+6)(x-6)}{\sqrt{(x-6)^2}}$, which simplifies ...
2
votes
1answer
27 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
101 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.
0
votes
1answer
39 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 $2$. ...
0
votes
2answers
46 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
84 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
61 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
23 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
48 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
37 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
57 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
35 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 ...
1
vote
1answer
32 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
49 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
55 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
41 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 ...
7
votes
3answers
128 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
66 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
209 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
26 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
50 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
79 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
28 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
26 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
31 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
33 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
72 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
26 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.
2
votes
1answer
26 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
73 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
37 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
40 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
30 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
80 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
34 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
31 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
69 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 ...
0
votes
1answer
25 views

Prove that a polynomial an irreducible g has no multiple root in C

I was looking at a question from Artin from Algebra which says that an irreducible polynomial g in F[x] where F is subfield of $\mathbb{C}$. So as per the proofs I have seen so far says as - ...
1
vote
1answer
29 views

Simplifying square roots?

How would I simplify $\sqrt{\frac{800}{3}}$ preferably by a factor tree? I know it simplifies into $\frac{20\sqrt{6}}{3}$. I just don't know the steps to get there. Help please?
1
vote
2answers
72 views

Find the limit of a Riemann Sum

The function is $f(x) = 1-x^2$. I'm stuck as I can't factor the expression in the last line to find the limit.
0
votes
0answers
29 views

How to factor $(1-(i^2/n)(1/n)$ to isolate $i^2$ and form a sigma identity?

given sigma from $i=1$ to $n$ of $(1-(i^2/n)^2)(1/n))$ how would you factor this function to isolate $i^2$ and get $[n(n+1)(2n+1)]/6$ ? update... I got until the limit as n approaches infinity (1/n) ...