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

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0
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3answers
30 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
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3answers
33 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 ...
1
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5answers
81 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 ...
0
votes
0answers
22 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 ...
5
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0answers
305 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
1
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0answers
31 views

Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications Example Prove that $\sqrt{2}$ is irrational by giving a proof by contradiction. Solution Let $p$ be the proposition "$\sqrt{2}$ is ...
3
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2answers
27 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: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ ...
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
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
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1answer
23 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 ...
2
votes
3answers
114 views

How to factor intricate polynomial $ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $

I would like to know how to factor the following polynomial. $$ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $$ What is the method I should use to factor it? If anyone could help.. Thanks in advance.
0
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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: $$ ...
4
votes
2answers
108 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 $?
1
vote
1answer
49 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
29 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
32 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. ...
0
votes
0answers
9 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 ...
0
votes
2answers
35 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
13 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: $P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$ ...
0
votes
3answers
98 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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0answers
51 views
1
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0answers
28 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
44 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
32 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
25 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
47 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?
-1
votes
4answers
102 views

How to factor $a^2-2a$

The part of the problem I'm doing has me factoring this: $a^2-2a$ and I'm at a loss on how to factor it. Would I be right in saying: $(a-1)^2$ Okay so I just ran across this part now: $a^4-16$ I'm ...
0
votes
1answer
67 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: ...
5
votes
4answers
388 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 $z=-\sqrt{1+6i}$ $\therefore$ ...
1
vote
1answer
20 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) ...
3
votes
1answer
777 views

Simplifying expressions - factoring or expanding?

The term "simplify" has always confused me. What does "simplifying" mean? More specifically, which is more "simple" - a fully factored expression or a fully expanded one?
1
vote
1answer
30 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. ...
7
votes
4answers
19k views

Find the largest prime factor

I just "solved" the third Project Euler problem: The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ? With this on Mathematica: ...
0
votes
2answers
48 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 ...
2
votes
4answers
73 views

Question on Factoring

I have very basic Question about factoring, we know that, $$x^2+2xy+y^2 = (x+y)^2$$ $$x^2-2xy+y^2 = (x-y)^2$$ But what will $$x^2-2xy-y^2 = ??$$ $$x^2+2xy-y^2 = ??$$
0
votes
1answer
13 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
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2answers
51 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
0answers
18 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
38 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
12 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 ...
0
votes
2answers
122 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
59 views

Irreducibility and factoring in $\mathbb Z[i], \mathbb Z[\sqrt{-3}]$

In $\mathbb Z[i]$, prove that $5$ is not irreducible. In $\mathbb Z[\sqrt{-3}]$, factor $4$ into irreducibles in two distinct ways. I am completely stumped on how to do this. I really need all ...
1
vote
2answers
79 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
33 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 ...
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, ...
5
votes
2answers
240 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 ...