For questions about finding factors of e.g. integers or polynomials
1
vote
3answers
46 views
find out the value of $\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$
If $(x-3)^2+(y-5)^2+(z-4)^2=0$,then find out the value of $$\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$$
just give hint to start solution.
13
votes
3answers
333 views
Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$
Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid?
I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
12
votes
2answers
92 views
Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$
How do I simplify:
$$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$
Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
5
votes
6answers
207 views
Cubing a simple thing
I am trying to do
$$(x + 2)^3 $$
I am actually not to sure what to do from here, the rules are confusing. To square something is simple, you just foil it. It is easy to memorize and execute. Here ...
2
votes
1answer
41 views
Find a divisor satisfying a given congruence
Suppose I have a highly composite positive integer $N$ with at least $10^{15}$ divisors for which I know the prime factorization.
Given $M$ with $\gcd(M,N)=1$ is there an efficient way to find a ...
2
votes
0answers
31 views
Find the factorization of the polynomial as a product of irreducible [duplicate]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$
Testing with the simplest possible root in this case, $P(1)=0$
Applying the ...
5
votes
1answer
181 views
Find the value of $x^3-x^{-3}$ given that $x^2+x^{-2} = 83$
If $x>1$ and $x^2+\dfrac {1}{x^2}=83$, find the value of the expression$$x^3-\dfrac {1}{x^3}$$
a) $764$
b) $750$
c) $756$
d) $760$
In this question from given I tried to ...
2
votes
2answers
43 views
Faulty velocity question?
If a ball is thrown vertically upward with a velocity of $160 \text{ ft/s}$, then its height after t seconds is $s = 160t − 16t^2$.
a) What is the velocity of the ball when it is $384 \text{ ...
4
votes
2answers
93 views
Irreducibility of $x^n-x-1$ over $\mathbb Q$
I want to prove that
$p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible.
My attempt.
GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
0
votes
1answer
20 views
Relationship between 2 Dimensional Quadratic systems and roots
Given four points
$(x_1, y_1)
(x_2, y_2)
(x_3, y_3)
(x_4, y_4)$
How does one construct a system of two equations:
$a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$
$b_1x + b_2x^2 + b_3y + b_4y^2 + ...
9
votes
5answers
77 views
Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$
Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed:
...
1
vote
3answers
109 views
Largest prime factor of 600851475143 [duplicate]
I'm trying to use a program to find the largest prime factor of 600851475143.
This is for Project Euler here: http://projecteuler.net/problem=3
I first attempted this with the code that goes through ...
1
vote
2answers
44 views
How do I factorize equations of the form $x^2 + Bxy + Cy^2 = 0$
Given equation
$$ x^2 + Bxy + Cy^2 = 0. $$
I want to factorize it in the form
$$ (x + my)(x + ny) = 0. $$
What are the values of $m$ and $n$ in terms of $B$ and $C$?
I tried writing the ...
5
votes
4answers
86 views
In order to factor we must find its zeros?
I am self-learning precalc (Precalculus Demystified) and found the following problem on page 170 :
Completely factor the the polynomial. $P(x) = x^3 - 5x^2 + 5x + 3; c = 3$ is a zero.
Since $c = 3$ ...
3
votes
3answers
66 views
Why all odd numbers not ending with 5 divide exactly into a number comprising only 9's?
Help me!!It's really frustrating I can't understand this simple thing.The maths instructor in my video,the renowned Arthur Benjamin,states (clip linked below) the following:
...
3
votes
3answers
316 views
Does knowing the totient of a number help factoring it? [duplicate]
Possible Duplicate:
Factoring a number $p^a q^b$ knowing its totient
Edit: The quoted question addresses only numbers of the form $p^a q^b$, I asked a general question for arbitrary $n$.
...
1
vote
3answers
64 views
Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$
For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$
where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
0
votes
1answer
47 views
How to simplify or factor this equation
$$1 = x + 2\cdot x$$
How can I simplify this formula for $x$.
1
vote
3answers
59 views
Solving quadratic equations by completing the square.
Graphing $y=ax^2+ bx + c$ by completing the square
Add and subtract the square of half the coefficent of $x$.
Group the perfect square trinomial.
Write the trinomial as a square of a ...
1
vote
4answers
70 views
I cannot find the last factor of this expression?
I'm supposed to factor $x^8-y^8$ (the exponents are 8 for both if it is too difficult to see) as completely as possible. It is easy to factor this to $(x+y)(x-y)(x^2+y^2)(x^4+y^4)$. However, the book ...
3
votes
4answers
113 views
Solving this 3-degree polynomial
I'm trying to factor the following polynomial by hand:
$-x^3 + 9x^2 - 24x + 20 = 0$
The simplest I could get is:
$-x^2(x-9) - 4(5x+5) = 0$
Any ideas on how I could go ahead and solve this by hand? ...
10
votes
3answers
465 views
Baby Shower Problem. Too hard for 1st grader but got parents thinking
So our six year old son comes home from 1st grade with the following math puzzle.
Your Aunt is having a baby. You have created a party game for a baby shower. It is called pick the gender. You ...
1
vote
1answer
34 views
Finding the possible lengths and widths, given a surface area.
Short Version of Question:
Each of $l$, $w$ and $k$ is a positive integer. Determine all possible values for $l$ and $w$ such that $l \ge w$, and $(k + 1)(l + w - 2k) = 133$.
Long Version of ...
1
vote
1answer
35 views
Foil, greatest common factor
I had this on a test and couldn't figure it out... It was written like this:
$12p^2-75$
The answer has to be $(p+\_ ) (p-\_)$
There must be a positive and negative number when combined (addition or ...
2
votes
3answers
63 views
How can I use prime factorization to find a cube root?
This is based on a lesson at Khan Academy that I didn't understand.
In the lesson, the instructor uses the number 512 as an example and the entire prime factorization consists of three groups of ...
0
votes
2answers
51 views
Simple question on factoring derivatives with “e”
I have a very simple factoring question; I'm doing a calculus problem in which part of the question requires me to factor a derivative. The derivative in question is $e^{-x}tx^{t-1}-e^{-x}x^t$ (the ...
2
votes
1answer
32 views
Creating and using calibration factors
Perhaps simple question, but I (the simple) need some guidance. The following applies to a project ongoing and is a challenge in that I am not a math whiz!
As example, I wish to measure temperature ...
4
votes
2answers
52 views
A set of numbers where none can be made by multiplying others in the set.
(I'm a programmer, please excuse my abuse of or lack of proper mathematical language)
The other day I needed to find a natural number that is cleanly divisible by all integers in the range ...
0
votes
3answers
59 views
How to factorize this?
$$x^2 + {16\over x^2} -12$$
This is what I've done so far...
$$x^4 -12x^2 +16$$
$$= (x^2 - 4)(x^2 + 4) -12x^2$$
What shall I do next?
EDIT: Thanks for the answer, however the answer given here is ...
3
votes
1answer
36 views
Factors of non-square
How do you solve to find how many of the positive factors of a number, say 36,000,000, are not perfect square? I know how to do this manually, which took me forever, but I want to be able to solve ...
7
votes
2answers
165 views
Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$
Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$
My try: $$\left(x^4-\frac{x}{2}\right)^2+\frac{3}{4}x^2-x+1 = ...
1
vote
4answers
63 views
How to factor $a^2-4b^2+4ac-8bc$
I am not sure how I would factor this. Can someone explain how I would factor this?
2
votes
4answers
58 views
How to factor $a^2+10ab+25b^2-9c^2$
This is what I got so far $a^2+10ab+(5b)^2-(3c)^2$. I think I can group $a^2+10ab$ but not sure about $(5b)^2-(3c)^2$. Can someone explain this?
1
vote
5answers
99 views
How to factor $x^4-7x^2-18$
I am not sure how I would factor this. The $x^4$ and $x^2$ are really throwing me off. Can someone explain how I would factor this?
2
votes
2answers
67 views
How do you factor $2ab-6ac-15c^2+5bc$
$$2ab-6ac-15c^2+5bc$$
Here is what I have done:
$$(2ab-6ac)-(15c^2+5bc)$$
$$2a(b-3c)-5c(3c+b)$$
$$2a(b-3c)+5c(-3c-b)$$
$$2a(b-3c)+5c(-b-3c)$$
$$(b-3c)(2a-5c)$$
I know this isn't the correct answer ...
0
votes
1answer
222 views
Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$?
Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$?
$Approach$:
$N$=$11^2$.$13^4$.$17^6$
$N^2$=$11^4$.$13^8$.$17^{12}$
This ...
1
vote
2answers
72 views
Factoring a long expression in the form $(a+b)^3 + (c - b)^3 - (c+b)^3$
I need to factor the following:$$\left(\dfrac{2}{3}x + \dfrac{5}{3}y\right)^3 + \left(\dfrac{3}{4}z -\dfrac{5}{3}y\right)^3 - \left(\dfrac{3}{4}z + \dfrac{2}{3}x\right)^3$$A friend of mine suggested ...
2
votes
2answers
47 views
Create a formula by given solutions
For my upcoming middle school exams I will need to convert a formula.
I have got the following question:
Create a formula which has the following solutions: $$ x_{1} = 5,\quad x_{2} = -3.$$
The ...
1
vote
2answers
62 views
What is “prime factorisation” of polynomials?
I have the following question:
Find the prime factorisation in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreduciblity in $\mathbb{Z}[x]$, of ...
5
votes
2answers
95 views
Factorize $a^2-ab-bc\pm c^2$
I got this question in a test but it did not specify the variable with respect to which I was supposed to factorize
$$a^2-ab-bc\pm c^2$$
where it could be just $a(a-b)-c(b\pm c)$ but no common ...
1
vote
2answers
57 views
Is $x^2+1$ irreducible over a cyclotomic field?
Let $K=\mathbb{Q}[\omega]$, where $1+\omega+\omega^2=0$, let $f(X)=X^2+1$. How can i prove irreducibility of $f$ over $K$?
2
votes
1answer
42 views
Can the Euclidean algorithm fail by not terminating in non Euclidean domains?
Is it possible for the Euclidean algorithm to fail by not terminating in finite time in non-Euclidean domains? In $\mathbb{Z}[X]$ it can fail by going out of the ring, ie one gets a non integer ...
2
votes
2answers
56 views
Is the polynomial $x^3 + 2x^2 + 1$ irreducible in $\mathbb{Z}_{17}[x]$?
Is the polynomial $x^3 + 2x^2 + 1$ irreducible in $\mathbb{Z}_{17}[x]$?
It seems this polynomial is reducible. How can I factor this? Thanks!
0
votes
3answers
71 views
Finding irreducible factors of a polynomial over $\mathbb{F}_3$
How to find irreducible factors of a polynomial? For example, how to find the irreducible factors of $x^4+1$ over integers mod 3? I have a start but need some help.
2
votes
2answers
74 views
Is there any known algorithm for factoring the fractional components of a binomial?
For a binomial such as $\binom {15} {6}=\frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1}$, it seems that it always divides evenly into an integer, and I ...
4
votes
3answers
122 views
How to factor $x^4 +3x -2$?
I have figured out there is two roots between $0$ and $1 ,-1$ and $-2$ for $x^4 +3x -2 = 0$.
Therefore there should be two factors $(x + a)$ and $(y - b)$ where $a,b \in R^+$.
But how to find these ...
3
votes
2answers
78 views
Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$
I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$.
I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
1
vote
2answers
39 views
Factoring into Linear Factors
How would one factor the following expression:
$(b - a)(c^2 - a^2) - (c-a)(b^2 - a^2)$
into the set of linear factors:
$(b - a)(c - a)(c - b)$
(This is not for homework but rather exam review. I ...
1
vote
2answers
51 views
Number of proper divisors generally prime
If we count the number of proper divisors of a positive integer, why do we usually get a prime number (or $1$)?
...
0
votes
0answers
38 views
How to find per second of failure?
I want someone to help me solve this.
An attacker is hitting the right username after 80% failed attempts
after every five minutes.
I got failure %, and time. What is the fail ratio how can i ...
