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

learn more… | top users | synonyms

0
votes
1answer
47 views

Use Fermat factorization to factor $809009\ldots$

Use Fermat factorization to factor $809009\ldots$ So far I have: \begin{align} \sqrt{809009} & = 889.449 \\ & = 890 \\[6pt] \sqrt{890^2 - 809009} & = 130\ldots ∉ \mathbb Z \\[6pt] \sqrt{...
5
votes
3answers
89 views

What's the best way to compute $\frac{a^4 + b^4 + c^4}{a^2 + b^2 + c^2}$

So, my teacher gave us this to compute yesterday, and I'm completly confused on how should I proceed : $$\frac{1^4 + 2012^4 +2013^4}{1^2 + 2012^2 + 2013^2}$$ I've tried several ways, but most of ...
0
votes
2answers
41 views

Factoring a 4th degree trinomial

I am trying to factor $3x^4-8x^3+16$, but I have no idea how to even start. I put into Wolfram Alpha, and it said that the answer was $(x-2)^2 (3 x^2+4 x+4)$. How would you factor something like this ...
2
votes
3answers
34 views

Find A and B for $A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$

Given:$$A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$$ How does one find A and B ? The answer is: $$A = 3x^{2};B=2x$$ but I can't see how one solves this. I tried subbing in values of x but it didn't lead ...
0
votes
0answers
23 views

Linear factors of minimal polynomial dividing $x^r$ - 1

I have a monic minimal polynomial $m(x)$ that divides $x^r - 1$. Apparently $m(x)$ has distinct linear factors over the complex numbers $\mathbb C[x]$. I understand this part, since $\mathbb C[x]$ is ...
3
votes
2answers
32 views

Find the cubic polynomial given linear reminders after division by quadratic polynomials?

A cubic polynomial gives remainders $(13x-2)$ and $(-1-7x)$ when divide by $x^2-x-3$ and $x^2-2x+5$ respectively. Find the polynomial. I have written this as: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ $P(x)=(...
0
votes
1answer
17 views

What is the theory of finding roots of a polynomial equation by looking at the factors of the $a_n$ and $a_0$ term called?

This is commonly taught in high schools in the context of factoring polynomials. I remember this method even has its own wikipedia page (with a proof) but I forget what was the theory called. Could ...
1
vote
1answer
25 views

What is the name of this kind of factoring algorithm

I just think about algorithm to find factor of number by doing something like guessing last digit of number and increase digit bit by bit Such as, I want to find factor of 749 Algorithm would begin ...
0
votes
3answers
23 views

Simplifying with exponents

So somehow I made it all the way to Calc II but struggle when it comes to this basic thing (not exactly the most 100% solid algebra foundation it seems). Unable to simplify this further: $$ =\frac{1}{...
1
vote
1answer
61 views

What will be the $(b^2-a)$

If $a$, $b$ are the real numbers and $$4a^{2}+b^2=4a-(\frac{1}{4b^2})$$ What will be the $b^2-a=$? I tried to be this equation more basic ,but i could not reach the result
1
vote
1answer
32 views

finding factors

How can i quickly find the factors of a particular number? Find the number of different factors of 1800 and 3003? This being the question , for 3003 i first found out its prime factors and then i ...
0
votes
1answer
33 views

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?
0
votes
1answer
26 views

What does the first x represent in {x, (x+1), (x-3)}?

The question is: "Part of the graph of a polynomial function is shown. Which of the following sets contains only elements that are factors of the polynomial?" The two answer choices left are B. {(x+1)...
0
votes
0answers
10 views

Which polynomial factorization method leads directly to $(1-\alpha_0 z)(1-\alpha_1 z)$

I know how to factor a polynomial $p(z)$ so that it looks like $a_n(z- z_0)\cdots(z-z_n)$, where $z_k$ are its zeros. Now I could squeeze this form into the wanted $(1-\alpha_0 z)\cdots(1-\alpha_n z)$,...
0
votes
2answers
40 views

Factor trinomials dividing by the common GCF

I have a doubt with the following problem I found in a book. You have to simplify a polynomial using the GCF. Now, this is the problem I am not able to grasp: $$6x^2-19x-7$$ According to the book, ...
0
votes
1answer
36 views

Simple question on factoring the difference of 2 perfect squares

(b) (i) Use the identity $A^2-B^2=(A-B)(A+B)$ to factorise the expression $5^{2k}-1$. Do I just put the k as 1 so that the equation is 5^2 and 1^2 Thanks Steve
0
votes
1answer
21 views

Solving a characteristic Polynomial of the Hilbert Matrix

I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation: $P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$ ...
1
vote
0answers
30 views

Factor polynomial series with coefficients of 0 or 1?

Is there any easy way to factor polynomials which have coefficients of only $0$ or $1$ and always have a $+1$ ? For example, factor $9^{25}+ 9^{19}+ 9^{14}+ 9^9 + 9^6 + 9^5 + 1$
2
votes
2answers
48 views

Can we use Eisenstein's Irreducibility Criterion to show that $x^4+1$ is not reducible in Q?

As such: Let $a(x)=x^4+1\in\mathbb{Q}\left[x\right]$. Then choose any prime $p$. By Eisenstein's Criterion, we see that $p\nmid 1$, $p\mid 0$ (since all coefficients of intermediate terms are 0), and ...
1
vote
1answer
35 views

Irreducible factorisation of polynomial over quotient field

Let $F=\mathbb{Z}_3[x]/<x^2+1>$. Factor $x^4+2$ into irreducibles in $F[x]$. I know that $F$ is a field since $x^2+1$ is irreducible. The usual way to find out that a polynomial is irreducible ...
0
votes
0answers
29 views

Finding Factors of a Determinant

Consider the determinant with elements: $a_{11} = ax-by-cz, a_{12}=ay+cz, a_{13}=cx+az$ $a_{21}=ay+bx, a_{22}=by-cz-ax, a_{23}=bz+cy$ $a_{31}=cx+az, a_{32}=bz+cy, a_{33}=cz-ax-by$ Where $a_{ij}$ ...
0
votes
1answer
49 views

How to factor $2x^4-11x^3-44x^2+149x+84$

I am doing something for math, and I need to factor $$2x^4-11x^3-44x^2+149x+84.$$ How do we factor it?
0
votes
1answer
74 views

General solution of the differential equation: y' cot x + y = 2

I have to find the general solution of the differential equation:$ y$' $cot$ $x$ + $y$ = $2$. And determine the integration constant using the initial condition $y$(0) = $1$. Additionally presenting ...
1
vote
1answer
17 views

Factoring a polynomial to get its zeros

While studying about sums and products of roots of polynomials, I found this on the web: We can take a polynomial, such as: $$f(x) = ax^4 + bx^3 +\dots$$ And then factor it like this: $$f(x)...
5
votes
4answers
395 views

Factorization of polynomials with degree higher than 2

I need help to factorize $x^4-x^2+16$. I have tried to take $x^4$ as $(x^2)^2$ and factorize it in the typical way of factorizing a quadratic expression but that did not help. Can someone help me to ...
0
votes
1answer
43 views

Simple question about finding roots of a polynomial

What am I doing wrong here? This is the denominator of one of my problems and I need to find the roots, so: $6i-z^2+1 \to z=\sqrt{1+6i}$ and $z=-\sqrt{1+6i}$ $\therefore$ $(z+\sqrt{1+6i})(z-\sqrt{...
0
votes
0answers
54 views

About an integer factoring algorithm

I have been toying with the following algorithm: ...
1
vote
1answer
34 views

Separable polynomials are the product of the minimal polynomials of their roots?

I see the following claim in this answer: Since $f$ is separable, it follows that $f(x)$ must be the product of minimal polynomials of [its roots] But, I don't know how we justify this claim. ...
0
votes
2answers
50 views

Properties of integer matrices $A$ such that $A^{p}=I$ for $p$ a prime integer.

Problem Statement: Let $p$ be an integer prime, and let $A$ be an $n\times n$ integer matrix such that $A^{p} = I$ but $A \neq I$. Prove that $n \geq p − 1$. We have been learning factoring of ...
1
vote
1answer
22 views

Factorization of vectors $(Z^T Z)^n Z^T-(X^T X)^n X^T$

Is there a way to factorize expression $(Z^T Z)^k Z^T-(X^T X)^k X^T$ where $Z$ and $X$ are real column vectors in $\mathbb{R}^n$, such that \begin{align} (Z^T Z)^k Z^T-(X^T X)^k X^T= (Z-X)^T P(Z,X) ...
1
vote
5answers
83 views

How to factorize a 4th degree polynomial?

I need help to factorise the following polynomial: $x^4 - 2x^3 + 8x^2 - 14x + 7$ The solution I need to reach is $(x-1)(x^3 - x^2 + 7x - 7)$. I need to factorize to this exactly as it is for a ...
0
votes
2answers
52 views

Simply factoring a quadratic equation

On pp 255 - 256 (footnote 7) of "Love & Math", Edward Frenkel states that we can factor a quadratic in terms of its solutions $x_1$ and $x_2$ as: $ax^2 + bx + c = a(x - x_1)(x - x_2)$ Where does ...
0
votes
1answer
14 views

Factoring binary quadratic form in two second order polynomials

I have a binary quadratic form in $N$ and $D$, $AD^2 + BND + CN^2$, where $A$, $B$, and $C$ are real coefficients and $N$ is a second order polynomial of $x$ with real roots $\lvert r \rvert <1$ ...
0
votes
0answers
20 views

Are all integral domains in which all irreducible elements are prime G.C.D domains?

I know that in G.C.D domains all irreducible elements are prime. Does the converse of this statement hold? If not, is there a weaker condition than being a G.C.D. domain that is both sufficient and ...
2
votes
0answers
52 views

Easy method of determining if a polynomial over $\Bbb{Z}$ has any quadratic factors with rational coefficients

There is an easy method of determining whether a monic polynomial $$\sum_0^n a_k x^k$$ with all $a_k \in \Bbb{Z}$ and $a_n = 1$ has any integer roots. At least it is easy if you can factor the ...
0
votes
1answer
13 views

Substitution of factors of the free term to find factors of a cubic equation

I was taught that finding a factor (and hence a solution) of a cubic equation may be easier if I try if the factors of the free term are roots of the equation. For example, if one has an equation $x^3-...
0
votes
2answers
123 views

$|x-2|$ as a factor of $|x^n-2^n|$ as a limit of function

I haven't posted in a while but I do have a question on factoring a specific term out of a particular polynomial function and I'm stuck at some point in the process. Please, let's see so that my ...
1
vote
2answers
41 views

Simple factor of equation

I have this polynomial: $5z^4-12z^3+30z^2-12z+5$ How do I factor it to get the following?: $(5z^2-2z+1)(z^2-2z+5)$ Can someone show me the procedure to perform whenever I encounter with a case like ...
1
vote
2answers
83 views

Factor 65 into irreducible in $\mathbb{Z}[i]$

Factor 65 into irreducible in $\mathbb{Z}[i]$ I tried to factor 65 in Gaussian integers by Mathematica, and I got $65 = -(1+2i)(2+i)(2+3i)(3+2i)$, but i don't know how to factor it by hand. Could you ...
2
votes
1answer
35 views

How to factor $n$-degree polynomials of this form?

I have come across a very specific form of polynomial, and I was hoping there would be a nice way to factor it - or at least show that it is irreducible. The form is $x^{n+1} + 2k \cdot x^n -1$ where $...
0
votes
1answer
60 views

Regarding the factorization $a^2+3b^2 = cd$.

Let $a,b,c,d$ be positive integers, with $\gcd(c,d)=1$, such that $$a^2+3b^2=cd.$$ By well-known classical results, we have that $c$ and $d$ are both of the form $u^2+3v^2$. QUESTION: Is it valid to ...
1
vote
4answers
51 views

Universal factoring method or list of methods (trinomials)

I am a student in calculus II. I'm now failing tests solely because I cannot factor; I understand everything else. This is compounded by the fact it seems to exceedingly hard to find anything ...
1
vote
4answers
60 views

How was this factoring of $1-2x-x^2$ achieved?

If I factor $1-2x-x^2$ using the quadratic formula I get $$x=\frac{2\pm \sqrt{4-4(-1)(1)}}{2(-1)}$$ $$x=\frac{2\pm \sqrt{8}}{-2}$$ $$x=-1 \pm \sqrt{2}$$ Let $\alpha = -1 +\sqrt{2}$ and $\beta=-1-\...
0
votes
0answers
36 views

Strange type of matrix equivalence, $\bf P=Q$. What applications or properties can it have?

Stemming from this question when actually searching for matrix similarities, having found this matrix equivalence: $$\bf A = PBP$$ That is neither transpose nor inversion on either of the $\bf P$s. ...
0
votes
0answers
25 views

Upper Bound on Number of Factors

Is there a theorem, lemma, or proof somewhere that proves an upper bound for the number of factors that a number can have? If not, would it be fairly trivial to prove that it is $log_2 n$?
-4
votes
1answer
74 views

Solve the surd equation [closed]

$$ \sqrt{14+ 8\sqrt{3}} = 2\sqrt{2} + b$$ Find b without using factoring b=sqrt{6}
-1
votes
2answers
44 views
0
votes
2answers
41 views

Is there a positive integer with $2010$ distinct positive integer factors?

Suppose that, $$f(n)=(\text{ the number of distinct positive integer factors of $n$ including $1$ and $n$ } )$$ Is there a positive integer $m$ such that $f(m)=2010$ ? How can conclude about the ...
12
votes
5answers
1k views

What is the sum of the reciprocal of all of the factors of a number?

Suppose I have some operation $f(n)$ that is given as $$f(n)=\sum_{k\ge1}\frac1{a_k}$$ Where $a_k$ is the $k$th factor of $n$. For example, $f(100)=\frac11+\frac12+\frac14+\frac15+\frac1{10}+\...
4
votes
2answers
122 views

How can we prove that a quadratic equation has at most 2 roots?

A quad equation can be factored into two factors containing $x $, but how can we prove that there no other sets of different factors yielding OTHER VALUES OF $X $?