Factoring Definition and 337 Threads

Why factoring difference of squares does not always work? For example ##(x^3 - x)(x^3 + x) = x^6 - x^2 ## but if I factor x^2 out from both I get ##(x^2(x-1))(x^2(x+1))## which is ##x^4(x^2 -1)=x^6 - x^4## Are there any rules I am not aware of? Tried using chat gpt and searching the web, but...

Hi, PF "Roots and Factors of Quadratic Polynomials There is a well-known formula for finding the roots of a quadratic polynomial. The Quadratic Formula The two solutions of the quadratic equation ##Ax^2+Bx+C=0##, where ##A##, ##B##, and ##C## are constants and ##A\neq{0}##, are given by...

Take the Klein-Gordon equation: ##\Box^2 = m^2## Say we want to linearize this equation, we try to come up with a new operator that squares into ##\Box^2##. ##(A\partial_t - B\partial_x - C\partial_y - D\partial_z)^2 = \Box^2## So we need ##-A^2=B^2=C^2=D^2=I## as this gives back the 2nd...

Dear Everybody, I have some trouble with this problem: Finding a sequence of elementary matrix for this matrix A. Let ##A=\begin{bmatrix} 4 & -1 \\ 3& -1\end{bmatrix}##. I first used the ##\frac{1}{4}R1##-> ##R1##. So the ##E_1=\begin{bmatrix} \frac{1}{4} & 0 \\ 0& 1\end{bmatrix}##. So the...

The summary abstract (describes the method) and full paper are linked. Summary abstract https://eprint.iacr.org/2021/232.pdf

I'm not sure if I should post this here or in the mathematics section. I'm trying to find a way to implement a mapping of a larger finite field such as GF(2^64) to a composite field GF((2^32)^2). Let f(x) be a primitive polynomial for GF(2^64), with 1 bit coefficients. If the coefficients of...

I do 7x^2-8x=15 7x^2-8x-15=0 I turn the problem into a Grouping(4 term) problem by multiplying 7*(-15) = 105 I find what numbers multiply to -105 and add to -8. -15 and 7 multiply to -105 and add to -8. 7x^2-15x+7x-105=0 I group them. 7x^2-15x | 7x-105=0 This is where i get lost, because i...

Is ##(r+y)^3## the same as ##[(1+y/r)^3*r^3]##. If not, how do I factor r out of the parenthesis.

https://arstechnica.com/science/2019/09/who-needs-qubits-factoring-algorithm-run-on-a-probabilistic-computer/

Hi, I'm currently learning QC with Nielsen's QCQI. I've written a program in Matlab following the factoring algorithm in page 233 and 235. I run the program for factoring 15, 18, and 25. I got the proper results several times, but I also got error reports sometime, especially when factoring 18...

Homework Statement Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}? Homework Equations The idea...

Hello, I have been going through the Wisconsin Placement Exam sample test. I'm trying to figure out how to find the solution set for x6 – 7x3 + 12. I have tried having u = x3 and solving for u2-7u+12, but I'm unsure what to do once I get (u - 4)(u - 3). Would someone help me figure out how to...

Homework Statement Factor ##x^4-3x^2+9## over the reals Homework EquationsThe Attempt at a Solution I am factoring this polynomial over the reals. So there are two options. It will either split into two linear factors and an irreducible quadratic, or two irreducible quadratics. I'm really not...

I have the simple quartic polynomial ##x^4+1##. How in general do I determine whether this is factorable over the reals or not? Since it has no real roots, it could only factor into two quadratic polynomials, but I am not sure what I can do to determine whether this is possible or not.

Homework Statement Solve for x: 3x^3 + 2x^2 + 75x - 50 = 0 The Attempt at a Solution I have tried substituting multiple values for "x" so that we get a factor, f(x)=0 I cannot seem to find an "x" value that will make this function=0. Is there a way to factor this function or did the book...

Homework Statement Completely factor: $$5abc^4-80ab$$ Homework Equations N/A The Attempt at a Solution $$5abc^4-80ab$$ $$5ab\left(c^4-16\right)$$ $$5ab\left(c^2+4\right)\left(c^2-4\right)$$ [/B] The correct solution is ##5ab\left(c^2+4\right)\left(c+2\right)\left(c-2\right)## I can see...

Methods of factoring . Method of common factors Factorization by regrouping terms Factorization using identities Factors of the form ( x + a) ( x + b) Factor by Splitting Is this all the factoring methods out there ? Or are there more ? I am also looking for a book with lots of...

Where can i find some worked out practice problems ?

Factor the expression. (a + b + c)^3 - a^3 - b^3 - c^3 I believe the best way to tackle this problem is to expand (a + b + c)^3. By doing so, I should be able to cancel out -a^3 - b^3 - c^3, right?

Factor the expression. (a - a^2)^3 + (a^2 - 1)^3 + (1 - a)^3 (a - a^2)(a - a^2)(a - a^2) + (a^2 - 1)(a^2 - 1)(a^2 - 1) + (1 - a)(1 - a)(1 -a) Is this the right approach thus far?

Factor. I need help getting me started with this monster question. [(x-a)^(-1/2)]/2 * (x+a)^(-1/2) - [(x+a)^(1/2)]/2 *(x-a)^(-3/2) Do I factor out (x - a)^(-3/2) as step one?

Factor (x^2 + 1)^(-2/3) + (x^2 + 1)^(-5/3) Solution: (x^2 + 1)^(-2/3)[1 + (x^2 + 1)^(2/5)] Yes?

Factor (x^2 + 1)^(3/2) + (x^2 + 1)^(7/2) Solution: (x^2 + 1)^(3/2)[1 + (x^2 + 1)^(7/12)] Correct?

Factor (5a^2 - 11a + 10)^2 - (4a^2 - 15a + 6)^2 Must I square both trinomials as step 1?

Factor (x^3/8) - (512/x^3). LCD = 8x^3 (x^6 - 8(512))/8x^3 (x^6 - 4096)/8x^3 [(x^3 - 64)(x^3 + 64)]/8x^3 In the numerator, the expression (x^3 - 64) is the difference of cubes, right?

Precalculus by David Cohen, 3rd Edition Chapter 1, Section 1.3. Question 50. Factor the expression. A^2 - B^2 + 16A + 64 Factor by grouping method. Group A = A^2 - B^2 Group A = (A - B)(A + B) Group B = 16A + 64 Group B = 16(A + 4) Group A + Group B (A - B)(A + B) + 16(A + 4) Correct?

If you write a polynomial as the product of two or more polynomials, you have factored the polynomial. Here is an example: how do i factor a polynomial ? can somebody explain this step in detail ?

A number can be factored into a product of its component factors A number can be factored into a product of its prime . But, What exactly is a prime number ? Prime numbers are numbers greater than 1 that are evenly divisible only by themselves and 1 Is it a number that can only be evenly...

Hi All, I am looking at some data regarding correlation between control measures and (perceived)errors at a given position. The level of error seems , intuitively, suspiciously low. I suspect the low perceived error may be at least partially explained by the Dunning Kruger effect...

I need to factor 6t^2+17t+7 when I break down the equation I get 6t^2+3t+14t+7 now I am not sure what to do after that to get my answer can someone please explain the steps for me thank you!

Homework Statement ##\frac{3x^2+x+C}{x+5}## find value of constant C such that the clause can be canceled in some manner. What will be the canceled form of the clause. Homework Equations -presumably C is a constant, and also an integer. -polynomial factorization will be attempted -cancelling...

i have no idea how to use the punctuation marks for writing equations..ill do my best.. (3a+4b)(a-3b) over b(3a+4b) multiplied by 1 over b^2(3b-a)(3b+a) during the cross cancelling the 3b-a would be multiplied by a negative one to switch the problem over right? (-1)(3b-a) in order to cross...

Homework Statement I just want to know how get from ##4x^3+3x^2-6x-5=0 ## to ##(x+1)^2(4x-5)=0##. What's the trick when dealing with these nasty polynomials? I got the answer by taking another approach (solving a root equation) but I noted this is also a way to go, but I can't figure out the...

Homework Statement 30(a^2+b^2+c^2+d^2)+68ab-75ac-156ad-61bc-100bd+87cd Homework EquationsThe Attempt at a Solution I've tried the form (Aa+Bb+Cc)(Da+Eb+Fc) expand this and equate to the orig expression I ended up getting (5a+3b-5c)(6a+10b-6c) which when I expand does not give me a correct...

Hi there, I need help with the following situation. Apologies if I'm not using the correct arithmetic terms! Variables: d,e,f e + f = g d / g = j j x e = K j x f = L K + L = M d = M the above situation is a simplified problem, which is easily solvable. Here's where I run into trouble...

Please assist me in this problem.$\frac{2}{3}b^5-\frac{1}{6}b^3+\frac{4}{9}b^2-1$ I tried grouping but still could not find anything factorable form of the expression. Regards.

I'm currently working through the book Algebra by I.M. Gelfand and A. Shen on my own. (As advised here: https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/#toggle-id-1) This isn't really homework so I wasn't quite sure where to post this. Anyway, I've got a question...

Hey (Wave) (Party), [FONT=Arial Black]Problem Factor $3x^2+11x+10$ [FONT=Arial Black]Workings This expression can be separated into, $3x^2+10x+x+10$ Where have i done wrong ? (Thinking) Many Thanks :)

I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or...

Is it possible to factor a quadratic equation along the lines of asin^2x -bsin2x+c ? If so, how? The sin2x seems to be a problem since when expanded it becomes 2sinxcosx, but I'm wondering if it is possible, and how it would be done? Thanks in advance.

Homework Statement Question: "If p and q are constants and x2+px+12 is equivalent to (x+3)(x+q), what is the value of p?" I was told that the answer was 7 but I am not sure how to reach this value myself. 2. Homework Equations I am not quite sure if this is applicable. The Attempt at a...

Homework Statement I did some refresher exercises in khan academy. I tried to wrap my head around the factoring problems, but it did not go well... I did not understand the khan academy tips and hints. (about factoring) one particular difficult polynomial was thus: factorize the polynomial...

I really hope someone here can help me - I'm sure that's not going to be difficult for most. :) I'm helping a friend's daughter factoring trinomials. I can get to the correct answer, but usually thru a system of trial and error and not a defined set of rules in every instance. Example 6x^2...

http://imgur.com/RNsBBoO (image) Can someone elaborate as to how he factored this? The textbook provided nothing else about factoring algebraic expressions with fractions than this image.It avoided fractions in factoring like a plague I guess. I understand the 3rd step to put the (x + 1)^(3/4)...

Homework Statement How to factor 2x^2 - (p-2)x - p ?? Homework Equations Basic factoring The Attempt at a Solution I don't know how to do it. The answer is 2(x-p/2)(x+2/2).. Help me please..

Hello! I just want to know the faster way to factor the expression. I already factored it out using trial and I am hoping you could give me some tricks to go about it faster than the usual method. Thanks $192x^3-164x^2-270x$

Homework Statement I am trying to factor a denominator so I can do a partial fraction decomposition to solve using a Laplace transform. denominator= 2s^3+3s^2-3s-2 Homework EquationsThe Attempt at a Solution

Since this is not available in my algebra textbook, how do you factorize the ##a^4 + b^4## and ##a^4 - b^4##? I also would like to know how do you obtain ##a^3 - b^3 = (a - b)(a^2 + ab + b^2)## and ##a^3 + b^3 = (a + b)(a^2 - ab + b^2)##?

Homework Statement 1. Let g(x) = x^4+46. a) Factor g(x) completely in ℚ[x]. b) Factor g(x) completely in ℝ[x]. c) Factor g(x) completely in ℂ[x]. 2. Completely factor the given polynomial in ℤ_5. [4]_5 x^3 + [2]_5 x^2 + x + [3]_5 Homework Equations ℚ = {m/n / m and n belong to Z, m is not...

Homework Statement ∫ (x2+2x)cos(x) dx The Attempt at a Solution I will not show the entire solution because it is quite long (had to use integration by parts twice), and my problem isn't with getting the solution. My problem is the book shows one answer (which is equivalent to mine, but...