Factoring Definition and 338 Threads

I know this is real simple stuff but can someone show me how you get xsquared -1 to be (x+1) (x-1)?

Homework Statement Factor x^3+3x+6 over Z5, Z10, Z, Q and ℝ. 2. The attempt at a solution For Z5, I have the roots of x^3+3x+1 in Z5, x=2=-3 and x=3=-4, so x+3 and x+4 are factors. By long division of x^3+3x+1, it is found that x+3 is a repeated factor, so x^3+3x+6=(x+4)(x+3)^2. For Z and...

For the life of me I cannot figure out how to factor higher order equations so that I can find the poles and zeros, my professor will not show how to do it and expects everyone to already know how, but i have forgotten and cannot find anywhere on the web to show a one and done method, please...

Homework Statement (1)Why can't I solve x^{4}-5x^{2}+4 in the following way: (x^{4}-4x^{2}+4)-x^{2} ... (x^{2}-2-x^{2})(x^{2}-2+x^{2}) ... If there is any reason why.. (2)How to solve it if the answer to get is (x-1)(x+1)(x-2)(x+2) ?

Hello, Homework Statement Factor: 4x^{4}-x^{2}-18 The Attempt at a Solution I solved a similar problem x^{4}-6x^{2}+9 by equating x^{2} to t and then reverse-FOIL'ing... this one just wouldn't give in... Completing the square also does not help to get the answer (presuming of course that...

Hello, this is rather complicated to explain so bear with me. I was wondering about the coefficients of polynomials which are factorable in the integers, meaning polynomials which can be written as (x+a)(x+b) where a and b are integers. I had a curious idea about letting the x-axis...

Homework Statement Ok, this is a pretty simple integral, but I'm having trouble with the factoring. \int \frac{1}{x^{2}+2} According to the book, the answer is: \frac{1}{\sqrt{2}} tan^{-1}(\frac{x}{\sqrt{2})} Homework Equations The Attempt at a Solution So I need to get it in the form of...

Homework Statement Trying to find relative min/max Homework Equations f(x,y)=X^4+y^4-36xy The Attempt at a Solution partial WRT(x) = 4x^3-36y partial WRT(y) = 4y^3-36x Set Partial WRT(x) = 0 4x^3-36y=0 4x^3=36y x=(9y)^1/3 plug x into Partial 4[(9y)^1/3]^3-36y=0...

As it turns out, this first part was an extremely pervasive user error that I have not seen for days. Still, though, it is interesting that equations can be factored in many different ways, so I will post that here instead: 3x^3 = -5x^2 +2x -3x^3 -5x^2 + 2x = 0 Factored: x(-3x+1)(x+2) and...

One of the homework questions i did tonight was (x^5-32)/(x-2), where i had to find the limit when x approaches two. In order to do this i looked up the formula to use for a difference of fifths and was able to solve the question by factoring out the denominator and putting two in for x, giving...

Homework Statement Factor X^6+8 Homework Equations The answer: (x^2 + 2)(x^4 - 2x^2 + 4) The Attempt at a Solution I'm really not sure how to arrive at the answer. Although, I have a feeling that I have to use (a + b)(a^2 - ab + b^2) because the question looks looks similar to...

Is there any kind of theorem, algorithm on factoring a large number into primes and which I could use (the theorem) with a 6-digit number? Thanks.

Two particles move along the x-axis. Particle one has the position x = 8t^2 + 7t + 2 Particle two has the acceleration a = -8t, and when t=0 v=23. When the velocity of the particles match what is their velocity? I thought of approaching this problem by changing both equations into the...

Homework Statement Give a formula for the extended function that is continuous at the indicated point. (x3-4x2-11x+30)/(x2-4), x=2 Homework Equations The Attempt at a Solution I know that I have to factor the top and bottom so that I can cancel terms that cause the function to...

I added an attachment and highlited my question... basically I am confused how they got there.

Homework Statement f(x)=6x^2+9x-6 Where does this function intersect the x-axis (i.e. what are the roots or zeroes of f(x))? Solve by factoring using the Box Method. When I use the method that is shown on this website http://www.purplemath.com/modules/factquad2.htm for this...

So let's say I have an equation x(x-3)2 = 0 I can factor it out to x(x-3)(x-3) = 0 Since there are two 3's in parenthesis, does this mean that the answers are x1 = 3 x2 = 3 and x3 = 0

Is there a difference between the form x^2 + ax + b and ax^2 + bx + c? I ask because I can use the AC factoring method for them both.

Homework Statement Factor x^{16}-x in F_8[x] Homework Equations The Attempt at a Solution I know how to do it in F_2[x] . I also feel the factorizations are the same in the two fields..but not sure how to prove it.

Hi all, I'm a beginner in calculus so my question might be stupid. When a function is differentiable, then in difference quotient one can always factor 'h' out in the numerator, even if the function is exponential and 'h' is in the exponent. Is some magic behind this or something else? I've read...

Homework Statement Consider the polynomial p(x) = x^5+45x^3+324x-3x^4 - 135x^2 - 972 . Given that p has some integer roots on the real and imaginary axes, factorise p into linear and quadratic factors with real coefficients. Enter your answer a set of factors in the form { x-1, x+4...

Sorry to ask help on such an easy problem. I'm self teaching, but for some reason I'm getting stuck on a step. Can anyone explain to me what to do once I reach the form below, prior to the answer? 1) 15(x-3)3 + 60x4(x-3)2 + 5(x-3) I get this far, as shown below. After this I'm...

Does anyone think that factoring will be shown to be "fractal" in nature, following the work of Ken Ono. Are there patterns in composite numbers, as yet unknown, that make factoring easier?

I have some questions about the presentation of the P vs NP problem here. I'm quite new to this stuff, so pardon my ignorance if I make dumb statements. http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf Specifically: First of all this statement is confusing to me...

Homework Statement Does x^3 + x^2 -1 factor? and if yes... how? Homework Equations The Attempt at a Solution

Hey, I'm a high school student (11th grade) and I'm working on a computer algebra system for a research project. Most things are are going well (sums, products, derivatives, integrals, series, expansion, complex analysis, factoring basic expressions, etc.). However, I am having difficulty...

Homework Statement I added attachment...

Homework Statement Consider the integral domain a + b \sqrt{10} . Show that every element can be factored into a product of irreducibles, but this factorization need not be unique. Homework Equations The Attempt at a Solution I know that this is not a unique factorization...

Homework Statement factor x^3 + x +2 = 0 Homework Equations I know the answer is (x-2)(x+1)^2 The Attempt at a Solution My question is when factoring problems like this - what should I look for or what should I group or separate to make the factoring easier...

Factoring a very long expression! please help (x^6) - (4x^5) + (x^4) + (10x^3) - (4x^2) - (8x) then: x((x^5) - (4x^4) + (x^3) + (10x^2) - (4x) - 8) my teacher skipped many steps and ultimately found: x * [(x+1)^2] * [(x-2)^3] I guess he's assuming we know how to get through it. Can...

Let T represent an odd target number to be factored. Let m represent any odd number > 1. If T is congruent to zero mod m, (T-1)/2 is congruent to (m-1)/2 mod m, and vice-versa. For example T 35 = 0 mod 5, 17 = 2 mod 5 77 = 0 mod 7, 38 = 3 mod 7 77 = 0 mod 11, 38 = 5 mod 11 47 = 0...

Homework Statement x^4 + 4x^3 - x^2 + 16x - 12 I know that with some higher order polynomials you can substitute say x^4 as a = x^2 thereby making it easier to break the thing apart and find its factors. I know I am looking for 4 roots, but my little substitution method doesn't really work...

say we have a number N=pq (known) and we only know the last digits of the two factors p and q ( the righter most digit of each factor ). Is-it possible to determine the other digits of the factors? If not, what is the minimum number of digits that should be known before the answer can be a yes?

Homework Statement lim t^2-9/t-3 x>3 Homework Equations The Attempt at a Solution I factored it into (t-3)(t+3)/(t-3) i then canceled out the (t-3)'s and substituted 3 to get 6 is this correct?

Homework Statement Could someone help me factor, ((x-1)y^2)+(1-(x^2)) The Attempt at a Solution Is it possible to get to: (x-1)(y^2-x-1)?

Homework Statement This is probably an easy question, but using the rational zero theorem I have not found any roots for this cubic polynomial. Factor the Following 6x^3-37x^2-8x+12 Homework Equations The Attempt at a Solution I have used all my knowledge of factoring and...

Homework Statement I need help trying to factor a trinomial. It has been a while, and I can't remember how to factor a trinomial with x^3. Please help. Sample problem... X^3 - 2X^2 + 1 Thanks Homework Equations The Attempt at a Solution Not sure where to start, I...

I need help trying to factor a trinomial. It has been a while, and I can't remember how to factor a trinomial with x^3. Please help. Sample problem... X^3 - 4X^2 + 5 Thanks

Homework Statement Factor the equation (without complex numbers) a^{10} + a^{5} + 1 This is a olympiad question The Attempt at a Solution I substituted a^{5} = x getting a quadratic eqation. But when I factored the quadratic equation I get complex roots and this is against the question...

I am taking an online Introductory Calculus course. I have a decent understanding thus far, however, the problem I'm working on gets somewhat messy and I am having a difficult time simplyifing the answer. f"(x) = (x^2 + 9)^2 (-2x) - [(9 - x^2)(2)(x^2 + 9)(2x)]/(x^2 + 9)^4 the solutions manual...

Homework Statement Factor this quadratic expression completely. 126x² - 15x - 66 Homework EquationsThe Attempt at a Solution 126x² - 15x - 66 (9x+6)(14x-11) I'm a good ways past this subject, but I find that it always takes me longer than some to complete these problems. If the coefficients...

Consider the multipole expansion of Newtonian potential 1/R in 2D in terms of Legendre functions. \begin{align} \mathbf{r_1} &= (r , \theta_1) \\ \mathbf{r} &= (r , \theta) \\ \phi(\mathbf{r}-\mathbf{r_1}) &= \frac{1}{\left|\mathbf{r}-\mathbf{r_1}\right|} \\...

Homework Statement Solve: x^3 - 9x^2 + 15x + 30 Homework Equations The Attempt at a Solution The factors of 30 are +-1, +-2, +-3, +-5, +-6, +-10, +-15, and +-30. I used my graphing calculator and got a zero close to -1. I plugged it into the original equation and got 5, not...

I know how to factor a quadratic where the x2 term is just x2 for example: to factor x2 + 5x + 6 I need to find 2 number that add to give 5 and multiply to give 6 (2 & 3). So the answer is (x + 2) (x + 3) However, I'm getting stuck with this one: 2x2 - x - 3 It seems that a different...

Homework Statement \Sigma2nn!/(n+2)! Homework Equations I'm using the ratio test because there are factorials but I'm a little stuck on whether or not to factor out The Attempt at a Solution lim 2n+1(n+1)!/(n+3)!*(n+2)!/2n(n)! After I set it up here I'm not sure of how to factor...

Does anybody know any tips on factoring large numbers. For example- x[2] - 70x + 901=0 I keep running into large numbers like these and spend a lot of time trying factor them out. Is their any short cut other than just trial and error? Thanks, EG

Homework Statement My problem is that whenever I have to use cubic functions - whether it's finding the roots (when dy/dx kis a cubic function) or finding the x-intercepts... With quadratic functions I usually use the quadratic formula. I try to figure out how to factor these equations in my...

This is not a homework question, per se, because I'm not a student. But it is a problem I found in a book. Actually, the problem doesn't involve what I'm going to ask, but it did present an opportunity for me to explore the subject. I have the graph of f(x)=x^{3}+3x^{2}-9x+3. I know the x...

Homework Statement Doing my vector mechanics dynamics homework and I cannot believe I am stuck on this part. trying to factor t^3 - 6t^2-36t - 40 = 0 Homework Equations The Attempt at a Solution I honestly do not remember where to begin. I remember there is a formula for perfect...

Homework Statement factor 4x^2-6xy+10y^2 Homework Equations The Attempt at a Solution completing the square i get, 4[(1/4)y^2-(3/4)xy+(9/16)y^2]-4(9/16)y^2+10y^2 but i don't know what to do with the bracketed term. It can't be factored easily