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

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-4
votes
2answers
131 views

Check if the number $3^{2015} - 2^{2015}$ is prime [closed]

Is $3^{2015} - 2^{2015}$ a prime. If not, why?
0
votes
1answer
15 views

What is the common factor in this case?

I have a sum as given below: $ -51y^7 - 34x^7 \over 17 $ What is the factored from of this question. Note : I know the factor of 51 and 34 is 17 but is the factor of -51 and -34 17 or -17????
1
vote
4answers
30 views

Complex trinomial factoring $ 2 \cos x - 2 = \sin^2 x$

$2 \cos x -2 = \sin^2 x$ I have been trying to solve this equation for the interval $0 \le x \le 2\pi$ . I figured I should keep them as one, so I put $2 \cos x -2 = 1 - \cos^{2} x$ however I ...
0
votes
2answers
30 views

How is the derivative of $x[4\sin(2x)+6\cos(2x)]$ the expression $(4-12x)\sin(2x)+(8x+6)\cos(2x)$

I am wondering because I have tried to answer this question, but have gotten a different answer: $(4-6x)\sin(2x)+(4x+6)\cos(2x)$. To get the above answer I did the following steps: 1) Product rule: ...
2
votes
2answers
47 views

Factorize : $(x+y+z)^p-[(-x+y+z)^p+(x-y+z)^p+(x+y-z)^p]$ where $p$ is an odd prime.

I am trying to factorize the expression: $$(x+y+z)^p-[(-x+y+z)^p+(x-y+z)^p+(x+y-z)^p]$$ where $p$ is an odd prime and $x,y,z$ are any non-zero integers. I know that it is divisible by $pxyz$. How do ...
0
votes
0answers
55 views

Factoring a quadratic polynomial, $4T^{2}-48T+144$

The question is asking me to factor the following polynomial to the simplest form. (without making it messy) \begin{align*} & 4T^{2}-48T+144\\ \end{align*} Here is how I do it but not sure which ...
0
votes
2answers
27 views

Packs of pencils required in equal

Stuck on the below question with my $9$-year old - any ideas on this one? Really don't know how to even start on this one..I know half of $72$ is $36$ but not too sure how that will help here...any ...
0
votes
1answer
19 views

Difference in the number of laps covered by two runners with different speeds

Stuck on this question for my 9-year old (no calculator allowed) - the section the teacher gave was in LCM and HCF - I think it is LCM to be used in this query? The LCM she worked out as 1200 but is ...
1
vote
2answers
61 views

tricks to find integer/prime factorization?

I am having a hard time when trying to find a prime factorization of a number. As an exempel $924=2^2 * 3 *7 *11$ Is there any shortcuts,trick,series of steps that leads to the prime factorization ? ...
0
votes
1answer
48 views

Completing the square with multiple variables

I'm trying to understand a solution to a PDE problem, and it involves reducing an expression by completing the square. I'm not sure how to go about the steps. The expression is: $$-x^2+2xy-y^2+4kty$$ ...
1
vote
1answer
50 views

How do I factor $x^8-x$ over $\mathbb{Z}_2$?

I am trying to factor the polynomial $x^8-x$ over $\mathbb{Z}_2$ to get a splitting field for it. I got that it is equal to $x(x-1)(x^3-x-1)(x^3+x^2+1)$ over $\mathbb{Z}_2$ but cannot proceed any ...
3
votes
2answers
28 views

Find all factored pairs of (a,b) such that…

Determine all possible ordered pairs (a, b) such that $a − b = 1$ $2a^2 + ab − 3b^2 = 22$ I've gone as far as factoring the left side of the second equation: $(a-b)(2a + 3b) = 22$ ...
1
vote
2answers
22 views

Simplify $(-2sin(t)-2sin(2t))^2+(2cos(t)-2cos(2t))^2$?

In calculating the length of a deltoid one gets the following string of trigonometric functions: $$(-2sin(t)-2sin(2t))^2+(2cos(t)-2cos(2t))^2$$ ...
1
vote
2answers
35 views

Finding the real zeros of $m(t)=t^7-3t^6+4t^3-t-1$

Finding the real zeros of $m(t)=t^7-3t^6+4t^3-t-1$. Using the rational theorem, I have found that $m(t)$ has $1$ as a zero with multiplicity of $2$. So $$m(t)=(t-1)^2(t^5-t^4-3t^3-5t^2-3t-1).$$ ...
2
votes
1answer
28 views

Unique factorization domain, problem with definition.

1) By my course, $\mathbb Z[t]$ is a unique factorisation domain. But I don't understand since $$10=2\cdot 5=(-2)\cdot (-5)=(-1)\cdot (2)\cdot (-5)=(-1)\cdot (-2)\cdot (-5)$$ which are different ...
3
votes
1answer
185 views

Quadratic Sieve

Can anyone explain how Quadratic Sieve (factorization algorithm) works? I tried reading relevant articles but they didn't include clear explanation / implementation of it.
-1
votes
1answer
42 views

How to factor the polynomial $24x^2y - 16x^3y^2$?

So for this question $24x^2y - 16x^3y^2$ I know the common factor between the two is 8 but Im not sure what to do next?
-2
votes
3answers
42 views

How to factor the polynomial $2x^2-7x-15$? [closed]

Having difficulties in factoring the expressions $2x^2 - 7x -15$.
1
vote
0answers
44 views

Reason for the method of factorization of cyclic expressions.

For example, I am given that factorize: $$a^2b+a^2c+ab^2+2abc+ac^2+b^2c+bc^2$$ So by the traditional method, we take the powers of $a$ $$=a^2(b+c)+a(b+c)^2+bc(b+c)$$ $$=(b+c)(a^2+ab+ac+bc)$$ ...
0
votes
1answer
29 views

Factoring the expression $3x+6y+x^2+2xy$

I need help with factoring the following expression: $3x+6y+x^2+2xy$ I am pretty much clueless as to how I need to approach this.
1
vote
2answers
32 views

When solving a simultaneous equation like this:

When solving a simultaneous equation like this: $2y - x = 4 $ $2x² + 3y² = x + 4y = 17 $ How do you express this second equation? I know how to solve simultaneous equations. I'm not just sure of ...
0
votes
1answer
24 views

Factor a given determinant using row and column operations. [duplicate]

When presented with the following: Use row or column operations to find the determinant in factored form: $\left\vert \begin{array}{llll} 1 & 1 & 1 & 1 \\ a & b & c & d ...
1
vote
1answer
26 views

Proving that $\sum_{j=0}^n 2^j=2^{n+1}-1$ for $n\geq 0$ by induction

Solving $S=$ $\displaystyle \sum_{j=0}^n$ $2$$^j$ = $2$$^n$$^+$$^1$ $-1$ So I was able to find the basis and the the RHS but I'm not sure how I should go about solving the LHS. Since I have K+1 in ...
1
vote
6answers
65 views

How do you factorize quadratics when the coefficient of $x^2 \gt 1$?

So I've figured out how to factor quadratics with just $x^2$, but now I'm kind of stuck again at this problem: $2x^2-x-3$ Can anyone help me?
0
votes
0answers
37 views

Algebra 2 Factoring Questions

Prove that the "sum of squares" is always a prime polynomial. Prove that all trinomials generated by the sum of cubes formula have no real roots. All the proofs online use concepts we haven't ...
1
vote
2answers
58 views

Decompose the polynomial $f(x)=\sum\limits_{n=0}^{100}x^n$ as product of irreducible polynomials

I'm trying to solve solve the next problem: Find all complex roots of the polynomial $f(x)=x^{101}-1$ Decompose the polynomial $f(x)=\sum\limits_{n=0}^{100}x^n$ as product of irreducible polynomials ...
2
votes
1answer
39 views

Find four integers summing to zero, with sum of cubes 24

I'm stuck on the following problem from Terence Tao's "Solving Mathematical Problems" Find all integers $a,b,c,d$ such that $a+b+c+d=0$ and $a^3+b^3+c^3+d^3=24$. (Hint: it is not hard to guess ...
1
vote
0answers
34 views

Factor out factorial from expression

I have the expression $(k+1)! - 1 + (k+1)(k+1)!$ How do I factor out $(k+1)!$ to achieve the result: $[(k+1)!(k+2)] - 1$? I for the life of me cannot figure this out. Thanks!
0
votes
1answer
42 views

Factoring quadratic equations

During the video on the link (at 20 seconds) the narrator says that $(x^2+3)$ cannot be factored, however I believe that it can be factored to $(x-1)(x-3)$ https://www.youtube.com/watch?v=4IeZkmO0STg ...
1
vote
2answers
11 views

System of Equations with Square Pattern

Find 16x + 25y + 36z if: x + 4y + 9z = 10 4x + 9y + 16z = 120 9x + 16y + 25z = 1230 I tried using "brute force" and solving for each variable but the numbers are very large and messy ( I do not want ...
2
votes
1answer
22 views

The transition from the residue classes modulo to the elements of classes

Let $K$ be an associative commutative ring with identity. Let $R$ be an ideal of the ring $K$. Consider the factor ring. Let $[\cdot]_1$, $[\cdot]_2$ and $[\cdot]_3$ - some residue classes that ...
1
vote
1answer
44 views

Statement from explanation of “What is the smallest number divisible by each of the numbers $1$ to$ 20?$” on project Euler

Here is part of explanation from the PE problem 5: Let us consider the case of finding the least value of $N$ for $k = 20$. We know that $N$ must be divisible by each of the primes, $p[i]$, less ...
1
vote
2answers
25 views

How do I factor this expression:$\frac{1}{64}x^3 - 8y^3 - \frac{3}{16}x^2y - \frac{3}{2}xy^2$

I need to f actor: $$\frac{1}{64}x^3 - 8y^3 - \frac{3}{16}x^2y - \frac{3}{2}xy^2$$ Tried to do it with an identity but failed, Factor theorem maybe ? Thanks
1
vote
3answers
47 views

Prove using factor theorem.

Using factor theorem, show that $a+b$,$b+c$ and $c+a$ are factors of $(a + b + c)^3$ - $(a^3 + b^3 + c^3)$ How do we go about solving this ? Thanks in advance !
4
votes
2answers
160 views

Show that $x^{n-1}+… +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime.

I proved that if $n$ is a prime, then $p(x)=x^{n-1}+\cdots+x+1$ is irreducible over $\mathbb Z$. But, I don't know how to prove that if $p(x)$ is irreducible over $\mathbb Z$, then $n$ is prime. Can ...
1
vote
1answer
28 views

Factorising Gaussian integers in general

Factorising $(1+3i)$ into the product of two Gaussian integers. So first I apply the G.I norm on this and obtain $\|1+3i\|=10=2\times 5$, so I expect the first Gaussian integer to have norm $2$ and ...
2
votes
4answers
48 views

Evaluating $x^4 + \frac{1}{x^4}$ given that $x^2 - 3x + 1 = 0$

Determine the value of $x^4 + \frac{1}{x^4}$ given that $x^2 - 3x + 1 = 0$. I've tried forcing in a difference of squares, looked for various difference of $n$s or sum of odd powers that I could ...
2
votes
2answers
32 views

Rewriting squareroot function in the form (a-b)(a+b)?

I have this function and I'm trying to write a program to compute it as n approaches 100. The problem is it overflows once it reaches around 50. The hint to solving this question is to rewrite the ...
2
votes
5answers
47 views

Is there an equation for factoring an quadratic equation.

Firstly, the title may be a little hard to understand so could someone please suggest a better one and make up for my 'ignorance.' Onto the question. If I have a quadratic equation: ...
0
votes
1answer
25 views

Any chances this can be further reduced?

I've come with the following equation, after a lot of simplification, but can't reduce further. Any chances it can be solved by reducing the $b$ and get the value of $a$? $$a = \frac{1000(1000 - ...
-1
votes
1answer
61 views

Factorization (Oxford MAT question) Help?

Oxford MAT test, Q3, please help: Suppose that the equation: $x^4 + Ax^2 + B = (x^2+ax+b)(x^2-ax+b)$ holds for all values of $x$ i. Find $A$ and $B$ in terms of $a$ and $b$. ii. Use this ...
2
votes
3answers
57 views

Factor theorem and polynomial solution

Find the value of $m$ if $(x-m)$ is a factor of $x^2-m^2 x+x+2$. I know if $(x-m)$ is a factor of $f (x)$ then $f(m)$ must be zero. But I could not reduce it.
1
vote
2answers
60 views

Factor $x^2+10x+15$?

How can you factor $x^2+10x+15$? The form $Ax^2+Bx+C$, where $B$ is the sum of $2$ factors of $C$ (and $\lvert A\rvert=1$) does not work.
2
votes
1answer
58 views

Factorization in a Group

$5$ is a prime number, but it can be expressed as $2*3*3$ mod $13$. So I am wondering if we are given a number $l$ and a prime $p$ that is smaller than $l$ but doesn't divide it, can we write write ...
0
votes
1answer
42 views

Which one is Faster: Factoring a polynomial of degree $c\cdot d$ or Factoring $c$ number of degree $d$ polynomials?

I consider polnomial $T(x)$ defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number (256-bit). I want to factorize the polynomials over the finit field. The dgree of $T(x)$ is ...
1
vote
2answers
67 views

The most Efficient Algorithm for Factoring Polynomial Over Finite Field

I have a polynomial defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 256-bit). The polynomail's degree is big (e.g. at least $10^5$). My Goal is: To find the ...
1
vote
0answers
15 views

How can I factorize $|z_1|^q z_1 - |z_2|^q z_2$?

Let $1\le q\le 2$. I would like to know that how I can factorize the following: $$ |z_1|^q z_1 - |z_2|^q z_2 = \\(z_1 - z_2) (\text{ a function of } z_1, z_2, \overline{z_1}, \overline{z_2}, q ) + ...
0
votes
1answer
19 views

I want to factorize as $a^qb - c^q d = (b-d)(\cdots \text{a function of }a,b,c,d\cdots)$

Let $2\le q\le 3$ and $a,b,c,d\ge 0$. I want to factorize the quantity as $$ a^qb - c^q d = (b-d)(\cdots \text{a function of }a,b,c,d, q\cdots). $$ Is this possible?
0
votes
1answer
19 views

Factoring vs dividing by $\mathbb Z$

Is $x|8$ the same as $x \equiv 8 / \mathbb Z$ when $x \in \mathbb Z$?
4
votes
7answers
160 views

If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?

If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$? At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?