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

learn more… | top users | synonyms

1
vote
3answers
80 views

Factorize $6x^2 -5x -14 = 0$

I'm throwing a bit of a blank on the best way to factor this : $$6x^2 -5x -14 = 0$$ I know that I could multiply $6$ by $14$ and then find a pair of factors that add to $-5$ (b), but this feels a ...
0
votes
1answer
40 views

Prove that a polynomial an irreducible g has no multiple root in C

I was looking at a question from Artin from Algebra which says that an irreducible polynomial g in F[x] where F is subfield of $\mathbb{C}$. So as per the proofs I have seen so far says as - ...
1
vote
1answer
37 views

Simplifying square roots?

How would I simplify $\sqrt{\frac{800}{3}}$ preferably by a factor tree? I know it simplifies into $\frac{20\sqrt{6}}{3}$. I just don't know the steps to get there. Help please?
2
votes
2answers
75 views

Find the limit of a Riemann Sum

The function is $f(x) = 1-x^2$. I'm stuck as I can't factor the expression in the last line to find the limit.
0
votes
0answers
42 views

How to factor $(1-(i^2/n)(1/n)$ to isolate $i^2$ and form a sigma identity?

given sigma from $i=1$ to $n$ of $(1-(i^2/n)^2)(1/n))$ how would you factor this function to isolate $i^2$ and get $[n(n+1)(2n+1)]/6$ ? update... I got until the limit as n approaches infinity (1/n) ...
0
votes
2answers
370 views

Notation: is a factor of

How can one write $x$ is a factor of $y$ (as a constraint)? I am also not sure what else to add to meet the question quality requirements.
0
votes
0answers
95 views

Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.

I have been going over this page as of late learning how to solve cubic formulas through depressing the equation, and solving for 'X'. Though, so far through numerous attempts, every single root I ...
1
vote
3answers
170 views

How do I factor $\ t^4-2 \ $?

This binomial is part of a bigger problem that I need to solve, however, I am little stuck on how to factor it. $(t-1)(t-1)(t+1)$ does not work.
0
votes
1answer
61 views

Factor Theorem given two factors

The function $f(x)= ax^3-x^2+bx-24$ has three factors. Two of these factors are $x-2$ and $x+4$. Determine the values of a and b and then solve for $f(x)$. Please give an algebraic solution.
4
votes
2answers
175 views

Number of Relatively Prime Factors

Given a number $n$, in how many ways can you choose two factors that are relatively prime to each other (that is, their greatest common divisor is 1)? Also, am I going in the correct direction by ...
0
votes
2answers
21 views

Can a logistic function(x+y) be approximately factored into $f_1(x)$ and $f_2(x)$?

I need to somehow factor the logistic function $$\frac{1}{1+e^{-(\theta-\beta)}}$$ into $f_1(\theta)$$f_2(\beta)$ approximately... namely $\frac{1}{1+e^{-(\theta-\beta)}} \approx ...
1
vote
4answers
200 views

For how many $n$, $x^6+n$ factors?

$\textbf{Question}.$ i) For how many integers $n$ with |$n$|$<500$, can the polynomial $p_n(x)=x^6+n$ be written as a product of two non-constant polynomials with integer coefficients? ii) How ...
4
votes
2answers
389 views

Definition of factor - Is n a factor of n?

Is there a universally agreed upon definition of what a factor of a number is? Is $n$ a factor of $n$? Is $1$ a factor of $n$? EDIT x 2: Integers Natural Numbers
0
votes
1answer
112 views

Factorizing Cubic Equations.

Factorization of Cubic Equations has always obstructed my way to the solution to a problem. Is there any simple technique to factorize them?
1
vote
2answers
49 views

Factoring Polynomial with Four Terms

To begin, I have the following equation: $f(x)=2x^2-5x$. Now I need to find the roots of $f(f(x))$. I also know that $f(x(x))$ simplifies to $8x^4-40x^3+40x^2+25x$. It's very obvious to me that a $x$ ...
0
votes
3answers
83 views

Rearranging the polynomial $x^3-23x^2+142x-120$ prior to factoring it

In the example 15: They are saying that, $$x^3-23x^2+142x-120 = x^3-x^2-22x^2+22x+120x-120$$ From where did $22x^2$ and $22x$ come and also $120x$. Please help me clear my confusion.
2
votes
4answers
26 views

Is $5^2x^3-x^5 = x^3(x-5)(x+5)$ or $-x^3(5-x)(5+x)$

Geogebra's Factor function says that $5^2x^3-x^5$ is $-x^3(x-5)(x+5)$ but from what I do, it is positive, $x^3(5+x)(5-x)$ Note the x isnt in the same position Am I wrong?
2
votes
3answers
110 views

How to factor $(x^5+1) (x^5-1) $

I have this: $ (x^5+1) (x^5-1) $ And I don't know how to continue factor. Geogebra's Factor says: $(x+1)(x-1)(x^4-x^3+x^2-x+1)(x^4+x^3+x^2+x+1)$
0
votes
1answer
32 views

factorization to determine convergence/divergence

Let: $$f(n) = n/(n^4+1)$$ if we factor f to: $$ n/[n^3(n+1/n^3)] $$ Given that this is equivalent, we perform distribution: $$1/[n^2(n+(1/n^3)]$$ We now have: $$ 1/(n^3+1/n) $$ So: $ 1/n ...
1
vote
2answers
59 views

finding solutions by factoring

How would you find the integer solutions to $a^2-b^2=16$? I know that the factors of $16$ are $8*2,$ $4*4,$ and $16*1.$ How would I use this? I know that $a^2-b^2=(a+b)(a-b)=16,$ but how would you ...
0
votes
0answers
29 views

How to decompose N into A and B so that A and B are close to each other.

I would like to decompose $N$ into $A$ and $B$ so that $A$ and $B$ are close to each other. Even I would allow some small error. Se I would like to have: $$N = A*B + \epsilon_N$$ where $$\epsilon_N ...
2
votes
1answer
33 views

factor $y'$ as $y' = f(x)g(y)$

I have a task where I have to write following differential equation as $y' = f(x)g(y)$ but I see no way you could factor it into two functions each only depending on $x$ respectively $y$: $$y' = ...
0
votes
0answers
220 views

Positive integers of sum and products

Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$ I just thought about this question and ...
0
votes
2answers
90 views

Pair of positive integers in product sums

I am still not sure on this answer. I would like someone to help me see the solution to his question. I was working on it for a while and it is the only question that I looked at that I can not ...
0
votes
1answer
109 views

Largest prime factor of a number

In Project Euler problem 3, where we have to find the largest prime factor of a number, one of the solution i came across is ...
1
vote
1answer
60 views

Break a number into factors.

I want to split a number in a product of factors, i am not sure how to do it. (2^48)+1 I think it should involve either mersenne or fermat little theorem. I have factored 48 and i have ...
0
votes
1answer
27 views

Factorizing $x^2-bxy+cy^2$

If $x^2-bx+c=(x+p)(x-q)$ ,then, factorise $x^2-bxy+cy^2$. My attempt - $(x+p)(x-q)=x^2+px-qx-pq$ $\implies p-q=b $ and $pq=c$ similarly-$p'-q'=by$ and $p'q'=cy^2$(assuming that on factorising ...
6
votes
1answer
110 views

factoring $x^n+x+1$

Is there a way of factoring a polynomial of the general form $$x^n+x+1$$ in the ring $\mathbb C[x]$ or $\mathbb R[x]$ or $\mathbb Z [x]$ for any $n \in \mathbb N$? (Or perhaps with certain conditions ...
0
votes
2answers
200 views

Factor $3x^{3/2}-9x^{1/2}+6x^{-1/2}$

I have $3x^{-1/2} (x^2-3x+2)$ However, I just tried to expand, and the answer is not the same as the original question. With fractional exponents I take out the smallest exponent, then I add the ...
0
votes
1answer
45 views

Polynomial Long Division in Algebra

How do i even begin to fathom these questions? How do i begin to answer them? Help would be much appreciated! Divide $X^5 - X^4 - 6X^3 - 8X^2 + 8X +48$ , by $X^2 - X - 6$. Hence fully factorise $X^5 ...
2
votes
1answer
64 views

Solve $x+\frac{2}{y}=3,y+\frac{2}{z}=3,z+\frac{2}{x}=3 $ in reals

Find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right. $$ Things I have done: first I ...
3
votes
3answers
77 views

Prove $\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$ if $\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$

if $a,b,c$ are real numbers and $$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0$$ Prove $$\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0$$ things i have done: using the assumption i ...
0
votes
5answers
103 views

How do I factor this? Simplifying for limit.

How do I factor $(s^5 - a^5)/(s^2 - a^2)$ ? I need to simplify it so I can find the limit as $s \rightarrow a$.
1
vote
3answers
51 views

Solve $ x^2+y^2=4, z^2+t^2=9, xt+yz=6 $ in integers

find answers of this system of equations in integers$$ \left\{ \begin{array}{c} x^2+y^2=4 \\ z^2+t^2=9 \\ xt+yz=6 \end{array} \right. $$ things I have done: we can observe that ...
2
votes
2answers
40 views

Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ if $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$

if $a,b,c,d$ are positive real numbers and $$a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$$ Prove $a^4+b^4+(a-b)^4=c^4+d^4+(c-d)^4$ Things i have done: from assumption $a^2+b^2+(a-b)^2=c^2+d^2+(c-d)^2$ I ...
1
vote
1answer
99 views

Factoring $a^4(b-c)+b^4(c-a)+c^4(a-b)$

I was solving the question that wanted to factor $a^4(b-c)+b^4(c-a)+c^4(a-b)$. My idea was to factor a $(c-a)$ in first step.So $$b(a^4-c^4)+ac(c^3-a^3)+b^4(c-a)=a^4(b-c)+b^4(c-a)+c^4(a-b)$$ ...
6
votes
5answers
94 views

Prove $a^2+b^2+c^2=\frac{6}{5}$ if $a+b+c=0$ and $a^3+b^3+c^3=a^5+b^5+c^5$

if $a,b,c$ are real numbers that $a\neq0,b\neq0,c\neq0$ and $a+b+c=0$ and $$a^3+b^3+c^3=a^5+b^5+c^5$$ Prove that $a^2+b^2+c^2=\frac{6}{5}$. Things I have done: $a+b+c=0$ So ...
3
votes
2answers
59 views

For which $p$ and $q$ polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $F_p[x]$?

It easy to prove that polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $\mathbb{Q}[x]$ if $(q,6)=1$, since they don't have a common zero in $\mathbb{C}$, this can be seen geometrically. My question ...
13
votes
10answers
4k views

Taking Calculus in a few days and I still don't know how to factorize quadratics

Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ...
1
vote
4answers
172 views

Factoring the following polynomials

Factorize the following polynomial: $t^3 -9t +8$ $t^6 -91t^2 +90$
3
votes
1answer
93 views

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
-1
votes
3answers
54 views

A problem on polynomial completely

$P(x)=x^3+mx^2+nx+14$ is divisible by $(x+2)$ but leaves a remainder of $-20$ when it is divided by $(x-2)$. Find the values of $m$ and $n$. Hence, factorise the polynomial completely. Now, I get ...
8
votes
2answers
625 views

Is 641 the Smallest Factor of any Composite Fermat Number?

Consider the sequence $a_n = 2^{2^n}+1$ of so-called Fermat numbers. It's well known that $a_5$ isn't prime ($a_5 = 641 \cdot 6700417$, this is due to Euler). What I want to know about this sequence ...
0
votes
2answers
96 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
3
votes
2answers
71 views

Factoring $x^{4} +1$, using real factoring to the second degree

Factoring to the second degree using real numbers $$x^{4} +1$$ I know that $ x^{4} +1=(x^{2} + i)(x^{2}-i).\;$ But these are complex but I thought using these in some kind of way? Got no where! ...
0
votes
1answer
235 views

Proof the Existence and Uniqueness of Factorization Form of Polynomial with Complex Coefficient

If $p\in P(\Bbb{C})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form $$p(z)=c(z-\lambda_1)....(z-\lambda_m)$$ where ...
2
votes
2answers
83 views

Using factoring to solve the equation $(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$

Solve for all values of $r$: $$(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$$ I'm not sure how my thinking isn't really correct here. I know this all seems very elementary and such, ...
0
votes
2answers
57 views

Factoring Questions

I have to complete a factoring packet for AP Calculus, and I'm having trouble with three of the questions... Find the missing factor: 1. $2\sqrt{x} + 6x^\frac 32 = 2\sqrt{x}$(_____________) ...
0
votes
2answers
92 views

Factoring the sum or difference of two cubes

I'm learning about sums and differences of cubes and I can't understand it very well. I am faced with this problem: $$x^3 - 27$$ I am told to find the sum or difference of the two cubes. I ...
0
votes
1answer
33 views

Factor complex equation

Im having some difficulty in factoring the following complex equation. The image bellow is taken from WolframAlpha, can anyone explain how I can factor this equation. In the task I am told one ...