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

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2answers
20 views

Factoring equations, non quadratic.

I'm taking the MIT opencouseware 6.042, Mathematics for CS. Working with induction proofs. It's been years since I've done this, and I'm not sure how he factored this. Assume p(n) true: $3|(n^3 ...
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1answer
31 views

What is the rule for simplying $ax^2 + bx + c$ for $a \neq 1$

I have a trivial quadratic $-3x^2 + 4x-4/3$ What is the most direct way to show it is equivalent to $-(3x - 2)^2/3$ Confusion: I used the quadratic formula and found the root to be $2/3$, and ...
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1answer
37 views

A polynomial with only one real $x$-intercept without imaginary roots has a root equal to $-c_n/(nc_{n-1})$.

Given coefficients $c_n, c_{n-1}, c_{n-2}, \ldots$ of the polynomial $c_n x^n+c_{n-1}x^{n-1} + \cdots +c_{1}x+c_0,$ prove that for $c_nx^n+c_{n-1}x^{n-1} + \cdots +c_1 x+c_0 = 0,$ $x=-c_n/(nc_{n-1})$. ...
1
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1answer
38 views

Quadratic Sieve Algorithm: Why is $(x − \lfloor \sqrt{n} \rfloor)^2 ≡ n ($mod $p)$?

If someone here understands the Quadratic Sieve Algorithm, I'm having trouble understanding why every prime $p$ in the factor base needs to a prime such that $n$ is a quadratic residue modulo $p$. It ...
1
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1answer
47 views

Find all solutions which satisfy the given conditions $m=9n^3+30n^2-9n$

Let $$ m=9n^3+30n^2-9n $$ where $n \in \mathbb{Q^{+}}$ and $m \in \mathbb{Z} $ . Find all solutions which satisfy the given conditions. I thought oh, there is probably an infinite amount ...
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0answers
13 views

Finding coefficients of some polynomials

Assume we know $f(x)\in\Bbb Z[x]$ of degree at least $4$ and we also know $r\in\Bbb Z$. How do we find $\alpha(x),\beta(x)\in\Bbb Z[x]$ of any degree and $g(x),h(x)\in\Bbb Z[x]$ such that $$f(x)=(g(x)...
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1answer
142 views

Factoring the expression $(\sqrt{x^2} -a)^2 + M = 0$

Where, M stands for all other terms in the equation. This is a typical format that you'll see when taking affine sections of an n-torus. I think I figured out how to do it correctly, without violating ...
1
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1answer
61 views

Find a factorization for $P(z)=z^5+z+1$ with $z \in \mathbb{C}$.

Find a factorization for $P(z)=z^5+z+1$ with $z \in \mathbb{C}$. I am a bit confused actually. Is anyone is able to give me a hint to solve the problem involving complex numbers? I think I can use ...
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1answer
18 views

Existence of Integer Solution to Quadratic Root

Example Given $91=(6(1)+1)(6(2)+1)=6^2(2)+6(3)+1$. If we wish to find a possible replacement for six we might try solving $2x^2+3x-90=0$. The quadratic equation gives $$x=\frac{-3 \pm 27}{4}$$ Is it ...
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0answers
20 views

Intrinsic Factorization with Modular Extension

The question here is has anyone seen a factorization algorithm similar to this? What is it called? Start with this $XY=N$. Suppose we know one non-trivial factorization of $N$, $X=x$ and $Y=y$. ...
0
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0answers
25 views

Digit Sums and The Euclidean Algorithm

The numbers that can be cast out for natural base $B$ seem to all be divisors of $B-1$, and length of cycles of differences in sequences of multiples of these divisors seem to be preserved when ...
1
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0answers
30 views

Factoring Algorithm Using Multilinear Forms

I was wondering if anyone could identify for me a known name for this algorithm, I would appreciate it. I will give an example. Let $N=961$. Then write $(6x+1)(6y+1)=6(160)+1$ Where $0<x\le y$ ...
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5answers
62 views

What does it mean to get rid of $x^{12}$ in the expression $ x^{12} (1-x^4)^6(1-x)^{-6} $ in the following context?

What does the auctor mean when he says that he gets rid of $x^{12}$ in the following context ? (...) For example,consider distributing $23$ toys among $6$ children such that no child gets more ...
2
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2answers
77 views

Factor $x^2+x+1+i$ in $\mathbb{C}[x]$.

Factor $x^2+x+1+i$ in $\mathbb{C}[x]$. I know the roots are $-i$ and $-1+i$, but I don't know how to go about factoring such polynomial. I tried using the quadratic formula, but I got stuck half way. ...
0
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2answers
54 views

Best Method for Factoring this Expression

I'm wondering if anyone has an opinion on the best way of factoring the following expression. As one can see it is quite complicated. $$(15x^2)(x^3+{4})^{4}(1-2x^{2})^{3}+(12x)(x^{3}+4)^{5}(1-2x^{2})^...
0
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1answer
18 views

I need help factoring 6q(7(6q)^2 +5)

Can someone please show me how to factor $6q(7(6q)^2 +5)$ to show that it is a multiple of 6? I'm working on a division algorithm problem, and I understand concepts of div alg but I really don't have ...
3
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5answers
86 views

What is the best way to solve an equation of the form $(f(x))^2-a(f(x))+b=x$?

On a math contest I was told to solve the equation $$(x^2-3x+1)^2-3(x^2-3x+1)+1=x$$ For this particular problem I simplified by letting $$a\equiv x^2-3x+1$$ Then I continued with $$a^2-3a+1-x=0$$...
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0answers
19 views

Is this product of two product factorizations correct?

I am working on an induction proof and would like to know whether this product equality is true: $$\big (\prod_{i=2}^n (\lambda_i-\lambda_1) \prod_{n\ge i > j \ge 1}(\lambda_i - \lambda_j)\big )$$ ...
0
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2answers
23 views

Factoring 4 terms with a difference of squares as two of the 4 terms

Can someone help show me step by step how to factor: $9x^2 - 24xy + 16y^2 - 81$ I see a difference of squares in the last two terms and am stuck at this stage (did I start wrong in doing the ...
0
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0answers
38 views

Why is $(3x+2)^n$ $\implies$ $(x-(-\frac{2}{3}))^n$ a valid modification?

Why can one write $(3x+2)^n$ as $(x-(-\frac{2}{3}))^n$? Does it depend on the domain of $x$? Why does the $n$ exponent not cause a problem?
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2answers
42 views

Finite fields: factorization of the trace function over the base field

Let $q$ be a prime power, and $m$ a positive integer. The trace function from $GF(q^m)$ to $GF(q)$ is defined to be the mapping $$Tr : GF(q^m) \rightarrow GF(q) $$ $$Tr(x) = x+x^q+x^{q^2}+\cdots+x^{q^{...
2
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1answer
20 views

Does this factor into a dot product

A hopefully easy question. I have this: $$c^2+(x^2+y^2+z^2)(v_x^2+v_y^2+v_z^2)-2c(x*v_x+y*v_y+z*v_z)$$ And I was wondering if this somehow factors into $$(c-f(\vec{r}, \vec{v}))^2 $$ where $$f(\...
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4answers
82 views

Express $-a^3 b + a^3 c+ab^3-a c^3-b^3c+bc^3$ as a product of linear factors.

Express $$-a^3 b + a^3 c+ab^3-a c^3-b^3c+bc^3$$ as a product of linear factors. I have tried rewriting the expression as: $$ab^3-a^3b + a^3c-ac^3 +bc^3-b^3c$$ $$= ab(b^2-a^2)+ac(a^2-c^2)+bc(c^2-...
0
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0answers
92 views

Chudnovsky binary splitting and factoring

In this article, a fast recursive formulation of the Chudnovsky pi formula using binary splitting is given. For $S(a,b)$: $$ m = (a + b) / 2 $$ $$P(a,b) = P(a,m) P(m,b)$$ $$Q(a,b) = Q(a,m) Q(m,b)$$ $...
5
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3answers
136 views

Solving $10x^4-13x^2+4=0$

I just came across a question in my paper that asks me to solve for $x$ in $10x^4-13x^2+4=0$ I've only learned how to factorize quadratics and the quadratic formula, but I'm not sure how to factorize ...
1
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1answer
32 views

what are all ordered pairs $(a,b)$ for which $17a^2-10ab+2b^2-4a+2=0$

what are all ordered pairs $(a,b)$ for which $17a^2-10ab+2b^2-4a+2=0$ I am having a hard time solving this problem. I started by plugging $a=0$ and $a=b=1$ but that doesn't help. So the answer is ...
0
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3answers
44 views

If $y+\frac{1}{y}=5$, find in the simplest form the values of $y^3+\frac{1}{y^3}$

If $y+\frac{1}{y}=5$, find in the simplest form the values of $y^3+\frac{1}{y^3}$ So I wrote: $$y+\frac{1}{y}=5$$ with a common denominator which is: $$\frac{y^2+1}{y}=5$$ Multiply $y$ to the ...
0
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1answer
25 views

How would you factor out/simplify something like this: $((13x)^2-2^{y+4})/(x^2-2^y)$?

How would you factor out/simplify something like this: $$\frac{(13x)^2-2^{y+4}}{x^2-2^y}\ ?$$
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2answers
139 views

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

Is $3^{2015} - 2^{2015}$ a prime. If not, why?
1
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1answer
16 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
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4answers
31 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 don'...
0
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2answers
34 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: $...
1
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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
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0answers
57 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
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2answers
30 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
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1answer
22 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 ...
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2answers
68 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
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1answer
117 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
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1answer
53 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
29 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: $(a-b)(2a + 3b) = 22$ $(...
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2answers
29 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: $$(-2sin(t)-2sin(2t))^2+(2cos(t)-2cos(2t))^2$$ $$=4(sin^2(2t)+2sin(t)sin(2t)+cos^2(2t)-2cos(t)cos(2t)+...
1
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2answers
36 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
35 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
198 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
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1answer
44 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
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3answers
48 views

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

Having difficulties in factoring the expressions $2x^2 - 7x -15$.
1
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0answers
47 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
30 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
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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
votes
1answer
32 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 \...