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

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1
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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
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1answer
26 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 ...
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1answer
27 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 ...
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6answers
67 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?
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2answers
59 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
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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 ...
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0answers
36 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!
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1answer
44 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 ...
0
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3answers
97 views

How to reduce a polynomial to $2$ real polynomials?

Can somebody explain me the principle and to show it on this example how to find $2$ real polynomials that their multiply is $p(x)$: $$p(x)=x^{4}+1$$
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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
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1answer
23 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 ...
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1answer
45 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 ...
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2answers
29 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
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3answers
48 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 !
5
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2answers
238 views

Show that $x^{n-1}+\cdots +x+1$ 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
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1answer
29 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
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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
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5answers
49 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: ...
1
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1answer
27 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 - ...
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1answer
65 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
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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.
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2answers
61 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
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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 ...
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1answer
44 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 ...
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2answers
76 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 ...
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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
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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?
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1answer
22 views

Factoring vs dividing by $\mathbb Z$

Is $x|8$ the same as $x \equiv 8 / \mathbb Z$ when $x \in \mathbb Z$?
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7answers
163 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 $x$?
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3answers
40 views

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

If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$? Please help, thanks.
2
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0answers
105 views

Finding roots and factors of multivariate polynomials

I know that in order to factor a one dimensional polynomial one can find the roots with some method, for instance a numerical newton method. Then one can systematically divide with $(variable-root)$ ...
3
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2answers
159 views

Help factorising a sixth degree polynomial [closed]

I have to factorise- $$x^6+5x^3+8$$Answer is $$(x^2−x+2)(x^4+x^3−x^2+2x+4)$$.I have also used factor theorem.Please help me.Thanks in advance.
2
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3answers
46 views

Express $y= x^{3} - x^{2} - 5x - 3$ in its fully factorised form

Don't know how to do this, please help. I have never done factorising cubic polynomials and don't know how to go about this
5
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1answer
75 views

Factoring $a^2+b^2$?

I remember there was a way to factor $a^2+b^2$ into something along the lines of $(a+\sqrt{a}+b)(a-\sqrt{a}+b)$ . I tried every combination o pluses and minuses for this form, but I couldn't get back ...
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1answer
26 views

Factorising polynomials 2

Hi I need help solving this question please, I'am in year 10: If $(x − p)$ is a factor of $mx^2 + nx + q$, show that $−2\sqrt{mq} \le n \le 2\sqrt{mq}$.
18
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1answer
248 views

Factoring $x^n+n$ over $\mathbb{Z}[x]$

I was working on some polynomial irreducibility problems, and eventually got to wondering which positive integers $n$ had the property that $p_n(x)=x^n+n$ is reducible over $\mathbb{Z}[x]$ i.e. can be ...
6
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1answer
65 views

Need help solving a bi-quadratic polynomial…

The polynomial to be factorised as a product of two factors is- $$x^4+3x^2+6x+10$$. I checked the solution in wolfram alpha to be- $$(x^2-2x+5)(x^2+2x+2)$$. I tried to factorise it by expressing it as ...
2
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1answer
42 views

Derivative of a Product of a Variable Number of Terms

Take a function that is the product of degree $k$, such as $$f_k (x)=\prod_{n=2}^{k+1} g\left(\frac xn \right), k\ge 1.$$ What methods would on use to find $f^\prime_k(x)$ with respect to $x$ in a ...
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1answer
23 views

Rearrangement of Parabola

I am attempting to show that the following expression - $2 [(P)x^2 + (P+Q)x + P)$ can be rewrriten as $2 (x+1)(Px + Q$)... but I have come to no help. I did manage to get to $Px (x+1) + Q(x+1)$ ...
6
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1answer
133 views

What are the roots of the polynomial $x^{3}+3x-2\pi$ $?$

By using Descartes's sign rule , I can tell this polynomial $$x^{3}+3x-2\pi$$ has one real root. But I want to know what that root is and what the factorization of it ...
0
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6answers
108 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 ...
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2answers
39 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
0
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1answer
86 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
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3answers
74 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
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0answers
19 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.
0
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1answer
37 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 ...
0
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0answers
58 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
116 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: ...
0
votes
1answer
54 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.