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

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0
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2answers
36 views

Factorisation of a polynomial [closed]

I have a polynomial $$t^4-4\lambda t^2-4t^2 $$ I need to give a real value to $\lambda$ such that i get 4 real roots.
0
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4answers
61 views

How do I compute the sum of 2 squares

if $x+y=a$ and $xy=b$, what does $x^3+y^3$ equal? I understand that $x^3+y^3=(x+y)(x^2-xy+y^2)$ but I don't see how I can figure out what $x^2$ or $y^2$ equals
1
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2answers
73 views

Factor irreducible polynomial in Z[x] and R[x]

I've got a couple of problems from an old exam in abstract algebra that I have difficulty in understanding. 1) Write the polynomial $2x^3 - 10$ as a product of irreducible elements in ...
1
vote
1answer
42 views

Factoring and Simplifying

I'm trying to do this problem, $$(4x + 1)^{15}\cdot\frac{1}{3}(12x - 5)^{-\frac{2}{3}}\cdot 12 + (12x - 5)^{\frac{1}{3}}\cdot15(4x + 1)^{14}\cdot 4$$ I've gotten down to, ...
3
votes
3answers
79 views

Factors of zero?

How many integer factors of 0 are there, and what are they? I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?
1
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1answer
39 views

Probability distribution of count of factors for all numbers

Is the following known? Define "factor count" as the number of prime factors of the number, minus 1. For example: Prime numbers have a factor count of 1-1 = 0 4 has a factor count of (2 and 2)-1 = ...
1
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1answer
41 views

How to give irreducible factorization in $\mathbb{Z}_5$?

I have this polynomial: $$f=x^5+2x^3+4x^2+x+4$$ How can i find the irreducible factorization(in $\mathbb{Z}_5$)?I can find the roots easily but thats not enough.
0
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1answer
28 views

Prove inequality of following type other than in induction method

I need to prove the following. $$(1^r + 2^r +\ldots + n^r)^n > (n^n)(n!)^r$$ where $r$ being a real number. I tried to solve it through induction method but it got complicated. How to solve this ...
0
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1answer
33 views

Factoring for Strong Induction for Fibonacci Sequence

Fibonacci: prove the following theorem: define the Fibonacci sequence $\left\{ a_n\right\}_{n=0}^{\infty}$ by $a_0=a_1=1$ and for integers $k>1$, $a_k=a_{k-1}+a_{k-2}$. Then, for each integer $n$, ...
14
votes
1answer
544 views

Is factoring polynomials easier than factoring integers? [duplicate]

I was reading the book Algebra: Chapter 0 , by Paolo Aluffi, and came across the following assertion, in page 290, Exercise 5.9: It is in fact much harder to factor integers than integers ...
1
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2answers
55 views

What are irreducible factors?

What are Irreducible factors? I have to solve this question: Find the irreducible factors of $x^4 + 5x^3 + 8x^2 + 9x + 10$ in ${\bf Z}_{11} [x]$. I couldn't find any websites that explained ...
2
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1answer
145 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
1
vote
1answer
49 views

Determine the values of x for which the linear approximation is accurate to within 0.1.

So I've got a function: $$\frac{1}{(1+2x)^4}$$ with its linear approximation: $$1-8x$$ For all values of $x$ where the linear approximation is accurate within $0.1$, then surely we subtract the ...
0
votes
2answers
161 views

How factor with square root

I have the following equation that I'm trying to factor, but I'm stuck at the end. $$\frac{zx^{-4}\sqrt{x}(yz^4)^3}{z^7xy}$$ $$\frac{\frac{1}{x^4}\sqrt{x}(yz^4)^3}{z^6xy}$$ ...
1
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3answers
28 views

Need help with basic factoring equation

I'm just trying to brush up on my factoring of quadratic equations. $$\frac1{x+3} + \frac1{x^2 + 5x +6}$$ $$\frac1{x+3} + \frac1{(x+2)(x+3)}$$ $$\frac{(x+2)(x+3) + (x+3)}{(x+2)(x+2)(x+3)}$$ Then ...
0
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2answers
27 views

Polynomial identity for a sum

If $$f(x) = \sum_{i=0}^{n}A_i x^i \quad \text{ and } \quad g(x) = \sum_{i=0}^{n}B_i x^i$$ are two degree $n$ polynomials, then we can say that the polynomial $$h(x) = \sum_{k=0}^{2n}C_k x^k \quad ...
-2
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3answers
141 views

An intutive way to think about odd and even numbers. [closed]

What is an intuitive way to think about odd and even numbers? And about divisibility also...
3
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1answer
80 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
2
votes
1answer
17 views

Factoring a series of Matricies

I have a graduate physics text which has an arbitrary matrix $M$. I am given the following $M^{0} + M^{1} + M^{2}...M^{n-1}$ and then it says this is equal to $(M^{n}-I)(M-I)^{-1}$, where $I$ is the ...
1
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3answers
42 views

(Factorization) How can I factorize this?

I'm not sure about how to factorise this. I'd appreciate some help. Thanks! $(12x-y)^2-(4x-3y)^2$.
0
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1answer
21 views

Show that a Polynomial has certain factorization

$P(x)$ is a polynomial in $x$ of degree $\leq n-1$. Show that $P(x)$ has $n-1$ distinct roots and thus has the factorization $$k\Pi_{i=2}^n(x-a_i)$$, where the constant $k$ is the coefficient of ...
0
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0answers
29 views

Proof for uniqueness for ideal multiplication

I am across the following question here: The uniqueness of a special maximal ideal factorization Let R be a domain, and let I be an ideal that is a product of distinct maximal ideals in two ways, ...
1
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0answers
88 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where p is prime integer and f is primitive integer polynomial that is irreducible ...
3
votes
1answer
136 views

Extra help on inequality

Someone very helpfully provided an answer to an inequality. See Hard Olympiad Inequality However I don't get part of their answer. How did they get the last factorization??? Thanks so much for any ...
1
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0answers
13 views

By what factor do winning chances increase based on total value?

Say I am entering 24/7 in endless sweepstakes, contests, giveaways, drawings, etc. Assuming each one I enter has no less than 1 in 1,000 chances, but no more than 1 in 1 million (and I enter at least ...
0
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3answers
40 views

Basic Algebraic Manipulation

How would I solve for $X$ in this instance? I can't figure out how to get the $X$ variables by themselves and the known values on the other side by themselves. $2(4-X)(4-X)+X = 3$
0
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1answer
24 views

How to simplifying and solving this polynomial?

I have a problem with simplifying the polynomial. In the first time, I see that this polynomial is quite simple, but when I'm trying, I realized that this polynomial isn't as easy as I saw. Here is ...
0
votes
1answer
43 views

Factor this equation [closed]

Can someone factor this for me? $(x^{\frac{n}3}-a^{\frac{n}3})$ I am stuck on it. Let n be any natural number.
0
votes
0answers
31 views

Is this factorization true for all $n$ in the natural numbers

I need to know if $x-a=(x^{\frac{n}3}-a^{\frac{n}3})(x^{\frac{n+1}3}+a^{\frac{n}3} x^{\frac{n}3}+a^{\frac{n+1}3})$ Is true. I know its true for $n=1$, is it true for all natural numbers though?
5
votes
3answers
236 views

Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$

I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right? If I can use ...
0
votes
4answers
25 views

What formula do I use for factoring these?

An elementary question, but I am having a lot of discrepancies identifying the correct formula to use, I can do more complex ones but not the simple ones if that makes sense. a) $8x^3 + 1$ b) $m^2 - ...
1
vote
2answers
39 views

Factoring Fully.

I am completely confused as to what to do, I don't understand how to factor with the brackets. $$42x^7(a+10)+60x^5(a+10)-24x^2(a+10)$$ Also state factoring used... Please and thank you. Steps?!?!
1
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2answers
84 views

Factoring $x^4 - x^2 + 1$

I'm interesting in finding the possible quadratic factorization of this polynomial: $x^4 - x^2 + 1$. My first idea was to do long division by $x^2+1$, but I did get a remainder, so I presume this ...
2
votes
1answer
47 views

Factorising a complex polynomial over C

If $f(z)=z^3+7z^2+16z+10$, find all factors of $f(z)$ over $C$. If I had at least one zero or factor I would be able to find the others, but I just don't know how to start.
1
vote
2answers
184 views

Solving inequalities with fractions with unfactorable polynomials

So I've been cracking my head open trying to solve this inequality: $$\frac {x+1}{2-x} \le \frac {x}{3+x}$$ I've been taught you have to put all factors to one side of the inequality (leaving zero ...
0
votes
2answers
169 views

Bezout's Identity for polynomials

Im working out a problem where I find out the GCD of two polynomials using Euclid's Algorithm, and then I need to use Bezout's Identity to make $\gcd(r,s)=ra+sb$ The question gives me $x^5+1$ and ...
0
votes
3answers
64 views

Please help me with a (simple?) “solve for x” problem.

I'm preparing for the GRE and was working through an old textbook (chapter on quadratic equations "completing the square," if that helps) and got stumped on $\displaystyle x^2 +{\frac{5x}{a}} + 6x^2 = ...
1
vote
5answers
51 views

Factor fully $98g^2+112g+32$ by decomposition

By looking at this question I understand it is a complex trinomial so do I just decompose it??I have multiplied 98 by 32 getting 3136, but I'm not quite sure what comes next.
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votes
2answers
67 views

Factor fully $625-(y-2)^2$

So far, I have used $(y-2)$ twice (multiplying both) because of the exponent being $2$. But, I need to factor and that's when I get confused. Please help!
0
votes
1answer
405 views

factor to find an algebraic expression for the length and width of the rectangle

the area of the rectangle is defined by $$ 6x^2+13x-28 $$ so far, i have decomposed the expression to get $$(3x-4)(2x+7)$$ but, now i need to find the length and width and that's where I have a bit of ...
2
votes
1answer
69 views

Determine 2 values of $k$ so that $36m^2+8m+k$ can be factored over the integers

So, I really need help with this, thank you very much for helping me. Anyway, I understand that $36m^2+8m+k$ is a complex trinomial and when factoring I should use $a^2+2ab+b^2=(a+b)^2$, but this is ...
1
vote
3answers
118 views

how do you factor $x^2 +kx+40$ over the integer

please please help me, I'm having a lot of troubles. I tried to use a^2+2ab+b^2 formula (like i was told) but that's where get lost. I understand that Factoring uses the opposite operation, but 40 ...
1
vote
2answers
94 views

Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
1
vote
1answer
68 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
2
votes
5answers
134 views

Derivation of factorization of $a^n-b^n$

How does one prove that: $$a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\dots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right)$$ Better yet, why is $a^n-b^n$ divisible by $a-b$? I would very much appreciate some ...
3
votes
2answers
60 views

Is $\gcd(n, \lfloor\sqrt{n}\rfloor!)$ a solution to the factoring problem?

The factoring problem: Factor $n=pq$ given only $n$ where $p$ and $q$ are primes and $0<p<\sqrt{n}<q<n$ I found that $$\gcd(n, \lfloor\sqrt{n}\rfloor!) = p$$ Would this be considered a ...
1
vote
0answers
40 views

On number of different factorizations over integers of a number field

Let $K$ be a finite field extension of the rational numbers and let $\mathcal{O}_K$ denote its ring of integers. If a rational integer $n$ factors into two distinct ways into irreducible elements in ...
2
votes
2answers
66 views

Show that $P(X) -X$ divides $P(P(X))-X$

Let $P$ be a polynomial in $R[X]$. Then show that $P(X) -X$ divides $P(P(X))-X$
0
votes
1answer
27 views

Why is it the case that the common factors of $x$, $y$ are also common factors of $x + y$

Why is it the case that the common factors of $x$, $y$ are also common factors of $x + y$? For example, $10$, $4$ share factors $2$, $1$. $10 + 4 = 14$ which has factors $2$, $1$ Similarly, $100$, ...
0
votes
1answer
54 views

Struggling to prove that if $n$ is a non zero integer, and $m > 0 \mid n$ then $m \leq |n|$

i need to prove that if $n$ is a non zero integer, and $m > 0$ and $m \mid n$ ($m$ divides $n$), then $m \le |n|$. I feel like i can do it by a combination of proof by contradiction and cases (ie ...