0
votes
1answer
52 views

Nonlinear system of equations / factoring two-variable cubic over $\mathbb{R}$

About halfway through a homework problem, I end up with a three-way identity: $$\frac{uw}{v+w} = \frac{uv}{u+w} = \frac{vw}{u+v}$$ (I say I end up with..., but this is the method suggested by my ...
3
votes
2answers
33 views

$a,b,c$ are three distinct natural numbers. Then how many ordered triplets $(a,b,c)$ will exist such that L.C.M (a,b,c) = 144.

Let $a,b,c$ be three distinct natural numbers. Then how many ordered triplets $(a,b,c)$ will exist such that L.C.M (a,b,c) = 144. Here's how I proceeded, 144=$(2^4)(3^2)$, so 144 has 15 factors(1 ...
1
vote
1answer
40 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, ...
0
votes
1answer
20 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
votes
1answer
88 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 ...
0
votes
2answers
57 views

How to factor cubics having no rational roots

$$-8x^3 +8x -3 = 0$$ I've already tried the possible roots of $\pm 1$ and $3$ using the rational roots test, but none of these help break it down into something more workable. How do I solve this ...
0
votes
4answers
58 views

Factoring with rational exponents

I'm not quite sure how to do this question. Every way that I tried doing it didn't yield an answer that is equivalent to the original question. $$(2x+1)^{2/3}-4(2x+1)^{-1/3}$$ When I tried doing ...
-3
votes
2answers
75 views

How to factor $x^3-x^2+x+3$ [closed]

I have a partial fractions integral with $x^3-x^2+x+3$ in the denominator. How do I factor this?
2
votes
1answer
55 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
0
votes
1answer
61 views

Please help me with factorisation

Is it possible to write $$64x^6-112x^4+56x^2-7$$ in linear factors? If so, what are they? (Finding it really difficult to ask this question!!)
2
votes
1answer
175 views

Factorizing $(x-1)(x-3)(x-5)(x-7)-64$

We need to factorize: $$(x-1)(x-3)(x-5)(x-7)-64$$ We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$. So this means that if the ...
1
vote
1answer
30 views

Computations question

a) Determine the prime factorizations of 3850 and 4125 b) Find the value of d = gcd(3850,4125) c) List all the positive divisors of d This is what I have so far. a) 3850: 11, 5, 5, 7, 2 4125: ...
1
vote
1answer
88 views

Number of divisors of a number

Is there any trick to find the number of divisors of any number? For e.g., a quick way to tell the number of divisors of 987655432 (chosen randomly)? EDIT: And it has to be done without prime ...
1
vote
2answers
183 views

Getting rid of the denominator of a polynomial

I'm tutoring a high school precalculus student; our current topic is the roots of higher order polynomials. The problem we're solving is: Find a polynomial with the roots $\frac23$, -1, and $(3 + ...
1
vote
1answer
41 views

Specific Annual Examination Marks

Steve has recently got his annual exam result.He has got upper than 690 out of 750.His obtained marks has odd number of factors.What is his obtained marks?
1
vote
2answers
120 views

Factoring Complex Trinomials

What is the answer for factoring: $$10r^2 - 31r + 15$$ I have tried to solve it. This was my prior attempt: $$10r^2 - 31r + 15\\ = (10r^2 - 25r) (-6r + 15)\\ = -5r(-2r+5) -3 (2r-5) $$
1
vote
2answers
79 views

Number of factors of a big number

what is the number of the factors of 884466000.how can I do this math without using factoring calculator?
0
votes
2answers
32 views

Factoring Expressions

I can't seem to factor this expression: $$2(2x^2-x)^2-3(2x^2-x)-9$$ So far, this is what I have done: $$(2x^2-x)(4x^2-2x-3)-9$$ I'm not sure what to do after this though, any hints?
2
votes
2answers
93 views

Factoring a given polynomial

I am trying to factor the polynomial $$(a-1)x^2 + a^2xy+(a+1)y^2.$$ The problem previous to it in the book uses the method of factoring a polynomial of the form $$ax^2 + bx +c$$ by inspection, and ...
0
votes
1answer
315 views

Factor $x^6 +5x^3 +8$

I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do. Help appreciated. Thanks.
5
votes
6answers
186 views

Factoring Quadratic Trinomials

I'm currently doing some homework, but I'm COMPLETELY stuck on one problem. I need to factor the following trinomial: $$5x^2+7xy+2y^2$$ How can I solve this problem? I have no idea what to do ...
3
votes
3answers
6k views

How to factor a four term polynomial without grouping?

$$2x^3 + 9x^2 +7x -6$$ This equation doesn't factor by grouping, and other than that I have no idea how to solve this problem. Will someone please help?
15
votes
4answers
714 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.
1
vote
1answer
66 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 ...
8
votes
2answers
389 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 = ...
2
votes
2answers
82 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 ...
2
votes
2answers
485 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 ...
4
votes
3answers
167 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
144 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 ...
10
votes
5answers
654 views

Factor $(a^2+2a)^2-2(a^2+2a)-3$ completely

I have this question that asks to factor this expression completely: $$(a^2+2a)^2-2(a^2+2a)-3$$ My working out: $$a^4+4a^3+4a^2-2a^2-4a-3$$ $$=a^4+4a^3+2a^2-4a-3$$ $$=a^2(a^2+4a-2)-4a-3$$ I am ...
6
votes
4answers
257 views

Factorize $3m^4-6m^3+14m^2-6m+11$

I have this expression: $3m^4-6m^3+14m^2-6m+11=0$ and I want to factorize it in $(m^2+1)(3m^2-6m+11)$. How can I do it? Thanks for any help!
1
vote
1answer
332 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 ...
0
votes
1answer
220 views

Zeroes in a 3x3 Matrix Determinant

My professor found the cubic roots of a 3x3 matrix by doing the following. I don't understand how step 2 came about and why he applied the same for step 4 on row 1 instead of row 2. Step 1: ...
1
vote
1answer
48 views

factorization of numbers with euclidian approach

Anyone know about this topic?. Factorization of numbers with euclidian approach I searching in the internet, but i couldnt find any source of this topic?. Some one can help me about this topic?. I ...
1
vote
1answer
398 views

Product of 3 integers is 72, find the 3 integers that give the smallest sum

Product of 3 integers a, b, c equals 72, where every factor is positive integer. Find the integers a, b, c with the smallest sum. It's easy to get the factors of 72 manually and see that the 3 ...
6
votes
4answers
1k views

Find all real solutions to $8x^3+27=0$

Find all real solutions to $8x^3+27=0$ $(a-b)^3=a^3-b^3=(a-b)(a^2+ab+b^2)$ $$(2x)^3-(-3)^3$$ $$(2x-(-3))\cdot ((2x)^2+(2x(-3))+(-3)^2)$$ $$(2x+3)(4x^2-6x+9)$$ Now, to find solutions you must set ...
2
votes
5answers
387 views

Factor $4x^3-8x^2-25x+50$ completely

Factor $4x^3-8x^2-25x+50$ completely The highest numbers you can take would be $1$, $2$, or $4$. Neither of those apply to all. So let's try the $x$! But the last term $50$ doesn't have an $x$ ...
1
vote
1answer
102 views

Factorize $f$ as product of irreducible factors in $\mathbb Z_5$

Let $f = 3x^3+2x^2+2x+3$, factorize $f$ as product of irreducible factors in $\mathbb Z_5$. First thing I've used the polynomial reminder theorem so to make the first factorization: ...
0
votes
2answers
72 views

Find monic grade 3 polynomial in $\mathbb Z_p[x]$ then factorize

Let $f = 15x^4+22x^3-x=0$ a polynomial in $\mathbb Z_p[x]$, find the first prime $p$ value that will make $f$ result in being grade 3 and monic. Then factorize $f$ in $\mathbb Z_3[x]$ as product of ...
0
votes
2answers
83 views

Find a prime number $p$ so that $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ is divided by $x-\overline{2}$ in $\mathbb Z_p$

Let $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ be defined in $\mathbb Z_p$. Find a prime number $p$ so that $f$ can be divided by $g = x-\overline{2}$, then factorize $f$ as ...
0
votes
3answers
193 views

Determine monic and degree 3 polynomial in $\mathbb Z_p$

I stumbled upon this kind of problem and I really can't get the hang of it. Will anyone please outline the way to solve it? Determine for which of the first $p > 0$ values the polynomial $f = ...
1
vote
4answers
1k views

How to factor the quadratic polynomial $2x^2-5xy-y^2$?

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
1
vote
2answers
116 views

Basic factoring problem.

I trying to work through a problem and have become stuck at the following equality: $ \sum_{n=1}^{100}{ n^2+n - 1 - (n-1)^2} = \sum_{n=1}^{100}{(3n - 2)}$ I can't quite get my head around the ...
1
vote
2answers
2k views

Notation : What is the meaning of the (mod n) in factoring algorithms?

Pretty much every thing is in the title, really! I'm trying to come up with an efficient algorithm to factorize large integer as an homework for a parallel programming course. I've seen a few pages ...
5
votes
2answers
174 views

elegant way to show $P= t^{1024}+t+1$ is reducible in $\mathbf{F}_{2}[t]$

This is homework exercise: $$P=t^{1024} + t + 1 , R = \mathbf{F}_{2}[t] \Rightarrow P \ \text{reducible in R}$$ I wanted to show this analogous to how a book shows it (book shows it with other ...
0
votes
3answers
115 views

finite fields factorization

Let $\mathbb{F}_2$ be the finite field with two elements. Let $f(x) = x^6+x^4+x+1$ be in $\mathbb{F}_2[x]$. If $f(x)$ is irreducible, give a reason. If it is not irreducible, determine a factorization ...
5
votes
6answers
7k views

How do you factor $x^3-3x^2+3x-1$?

$$x^3-3x^2+3x-1?$$ I know this may seem trivial, but I, for the life of me, I cannot figure out how to factor this polynomial, I know that the root is $$(x-1)^3=0$$ because of wolframalpha, but I ...
2
votes
3answers
191 views

How to factor $2x^2 - 8y^2$

How to factor $2x^2 - 8y^2$ ? So far I got it down to $$2(x^2 - 4y^2),$$ but it's not the answer; I don't think it's factored enough.
0
votes
2answers
953 views

Factor By Grouping 3rd Degree Polynomial

Just to be upfront, this is a homework question, I already know the answer, but I can't figure out how to get there or the logic behind the hint, which is really what I'm after. Please don't solve it ...
1
vote
3answers
139 views

Trigonometric factoring

Very next question, no idea what to do... I am suppose to factor $2\sin^2x + 3\sin x+1$ . I figure this is pretty simple so I do $(2\sin x)(2\sin x)+3 \sin x+1$ . For some reason this is incorrect ...