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

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3answers
141 views

Factoring of Degree 4

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. And every method that I have used so far from simply ...
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0answers
20 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 ...
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0answers
21 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 + ...
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4answers
65 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 ...
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2answers
22 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 * ...
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1answer
14 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 ...
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1answer
14 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: ...
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1answer
30 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 ...
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3answers
63 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.
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0answers
15 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.
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1answer
17 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 = ...
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2answers
35 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.
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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?
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0answers
21 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 ...
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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)?
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3answers
71 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 ...
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2answers
24 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 ...
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1answer
24 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 ...
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3answers
65 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 ...
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1answer
16 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 - ...
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1answer
27 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?
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2answers
57 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.
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0answers
22 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) ...
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2answers
30 views

Notation: is a factor of

How can one write $x$ is a factor of $y$ (as a constraint)? I am also not sure what else to add to meet the question quality requirements.
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0answers
54 views

Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.

I have been going over this page as of late learning how to solve cubic formulas through depressing the equation, and solving for 'X'. Though, so far through numerous attempts, every single root I ...
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3answers
50 views

How do I factor $\ t^4-2 \ $?

This binomial is part of a bigger problem that I need to solve, however, I am little stuck on how to factor it. $(t-1)(t-1)(t+1)$ does not work.
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1answer
35 views

Factor Theorem given two factors

The function $f(x)= ax^3-x^2+bx-24$ has three factors. Two of these factors are $x-2$ and $x+4$. Determine the values of a and b and then solve for $f(x)$. Please give an algebraic solution.
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2answers
52 views

Number of Relatively Prime Factors

Given a number $n$, in how many ways can you choose two factors that are relatively prime to each other (that is, their greatest common divisor is 1)? Also, am I going in the correct direction by ...
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2answers
20 views

Can a logistic function(x+y) be approximately factored into $f_1(x)$ and $f_2(x)$?

I need to somehow factor the logistic function $$\frac{1}{1+e^{-(\theta-\beta)}}$$ into $f_1(\theta)$$f_2(\beta)$ approximately... namely $\frac{1}{1+e^{-(\theta-\beta)}} \approx ...
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4answers
99 views

For how many $n$, $x^6+n$ factors?

$\textbf{Question}.$ i) For how many integers $n$ with |$n$|$<500$, can the polynomial $p_n(x)=x^6+n$ be written as a product of two non-constant polynomials with integer coefficients? ii) How ...
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2answers
287 views

Definition of factor - Is n a factor of n?

Is there a universally agreed upon definition of what a factor of a number is? Is $n$ a factor of $n$? Is $1$ a factor of $n$? EDIT x 2: Integers Natural Numbers
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1answer
43 views

Factorizing Cubic Equations.

Factorization of Cubic Equations has always obstructed my way to the solution to a problem. Is there any simple technique to factorize them?
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2answers
30 views

Factoring Polynomial with Four Terms

To begin, I have the following equation: $f(x)=2x^2-5x$. Now I need to find the roots of $f(f(x))$. I also know that $f(x(x))$ simplifies to $8x^4-40x^3+40x^2+25x$. It's very obvious to me that a $x$ ...
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3answers
68 views

Rearranging the polynomial $x^3-23x^2+142x-120$ prior to factoring it

In the example 15: They are saying that, $$x^3-23x^2+142x-120 = x^3-x^2-22x^2+22x+120x-120$$ From where did $22x^2$ and $22x$ come and also $120x$. Please help me clear my confusion.
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4answers
25 views

Is $5^2x^3-x^5 = x^3(x-5)(x+5)$ or $-x^3(5-x)(5+x)$

Geogebra's Factor function says that $5^2x^3-x^5$ is $-x^3(x-5)(x+5)$ but from what I do, it is positive, $x^3(5+x)(5-x)$ Note the x isnt in the same position Am I wrong?
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3answers
56 views

How to factor $(x^5+1) (x^5-1) $

I have this: $ (x^5+1) (x^5-1) $ And I don't know how to continue factor. Geogebra's Factor says: $(x+1)(x-1)(x^4-x^3+x^2-x+1)(x^4+x^3+x^2+x+1)$
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1answer
26 views

factorization to determine convergence/divergence

Let: $$f(n) = n/(n^4+1)$$ if we factor f to: $$ n/[n^3(n+1/n^3)] $$ Given that this is equivalent, we perform distribution: $$1/[n^2(n+(1/n^3)]$$ We now have: $$ 1/(n^3+1/n) $$ So: $ 1/n ...
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2answers
47 views

finding solutions by factoring

How would you find the integer solutions to $a^2-b^2=16$? I know that the factors of $16$ are $8*2,$ $4*4,$ and $16*1.$ How would I use this? I know that $a^2-b^2=(a+b)(a-b)=16,$ but how would you ...
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0answers
23 views

How to decompose N into A and B so that A and B are close to each other.

I would like to decompose $N$ into $A$ and $B$ so that $A$ and $B$ are close to each other. Even I would allow some small error. Se I would like to have: $$N = A*B + \epsilon_N$$ where $$\epsilon_N ...
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1answer
28 views

factor $y'$ as $y' = f(x)g(y)$

I have a task where I have to write following differential equation as $y' = f(x)g(y)$ but I see no way you could factor it into two functions each only depending on $x$ respectively $y$: $$y' = ...
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0answers
50 views

Positive integers of sum and products

Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$ I just thought about this question and ...
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2answers
57 views

Pair of positive integers in product sums

I am still not sure on this answer. I would like someone to help me see the solution to his question. I was working on it for a while and it is the only question that I looked at that I can not ...
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1answer
48 views

Largest prime factor of a number

In Project Euler problem 3, where we have to find the largest prime factor of a number, one of the solution i came across is ...
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1answer
32 views

Break a number into factors.

I want to split a number in a product of factors, i am not sure how to do it. (2^48)+1 I think it should involve either mersenne or fermat little theorem. I have factored 48 and i have ...
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1answer
24 views

Factorizing $x^2-bxy+cy^2$

If $x^2-bx+c=(x+p)(x-q)$ ,then, factorise $x^2-bxy+cy^2$. My attempt - $(x+p)(x-q)=x^2+px-qx-pq$ $\implies p-q=b $ and $pq=c$ similarly-$p'-q'=by$ and $p'q'=cy^2$(assuming that on factorising ...
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1answer
95 views

factoring $x^n+x+1$

Is there a way of factoring a polynomial of the general form $$x^n+x+1$$ in the ring $\mathbb C[x]$ or $\mathbb R[x]$ or $\mathbb Z [x]$ for any $n \in \mathbb N$? (Or perhaps with certain conditions ...
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2answers
108 views

How to find factors of very big number

I need to find factors of very big number say ($10^{1000}$). I.e if input is $100$ then output should be $10, 10$ because $10\cdot 10=100$. This is very simple if $N\le $ size of (long) but I want to ...
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2answers
72 views

Factor $3x^{3/2}-9x^{1/2}+6x^{-1/2}$

I have $3x^{-1/2} (x^2-3x+2)$ However, I just tried to expand, and the answer is not the same as the original question. With fractional exponents I take out the smallest exponent, then I add the ...
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1answer
18 views

Polynomial Long Division in Algebra

How do i even begin to fathom these questions? How do i begin to answer them? Help would be much appreciated! Divide $X^5 - X^4 - 6X^3 - 8X^2 + 8X +48$ , by $X^2 - X - 6$. Hence fully factorise $X^5 ...
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1answer
44 views

Solve $x+\frac{2}{y}=3,y+\frac{2}{z}=3,z+\frac{2}{x}=3 $ in reals

find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right. $$ Things i have done: first i ...