Factoring Definition and 337 Threads

Description of the Method We are given a trinomial of the form $ax^{2}+bx+c$, and asked to factor it into a product of two dissimilar binomials $(fx+u)(gx+v)$. The method that follows assumes $a,b,c$ have no common factor; if they do, you must factor out the greatest common factor before...

Homework Statement Let E denote the set of positive even integers. An element p ∈ E is called an E-prime if p cannot be written as a product of two elements of E. Determine a simple criteria for when elements of E can be uniquely factored into a product of E-primes. Homework Equations Some...

Hey Guys, So I'm currently factoring polynomials. As you all know,when trying to factor a polynomial with 4 terms,the answer can be the difference of a trinomial and monomial squares. So I had the problem 4a^2-4ab+b^2-c^2 I thought(book doesn't explain it well) that your first step was to...

I would like to know if it is possible to determine if a polynomial has rational zeroes, or, in other words, is unfactorable using whole numbers. For example 4x^3+2x^2-4x+25. I know you can use trial and error to sub in the factors of 25, and I understand the rational root theorem. However, I...

Can any polynomial in any degree of x be factored into a product of the form (leading coefficient)(x-a)(x-b) ... (x-z) as long as we can use complex numbers for a,b, etc.? Thanks

how do you convert this [-s(s-(s-1)-2s+n(n+1)] into (n-s)(n+s+1)

The Question: My Attempt: I use the 'Double Angle Formula' to get to the second line. Then multiply the sigma-x into what's in the brackets, then factor out at the end. I'm not sure how I reach the answer that's provided with the solution. Any help would be appreciated. Thank You.

So given the problem 2(x+3)(x-2)/(x-2)≥0 How come we can't factor x-2 in the numerator against x-2 in the denominator? I mean theyre all just factors right? Does it have something to do about it being an inequality? We got this problem on a test yesterday, and the correct way to solve it was to...

Homework Statement Hello! I understand that this is a very simple thing, but somehow I can't find the key :) Please, take a look a pictures attached with a problem and an answer. The task is to create a polynomial f with real number coefficients which has all of the desired characteristics...

Homework Statement Solve for x. a) 2x^3-3x^2-5x+6=0 Homework Equations -Find possible values of x -Divide that factor by 2x^3-3x^2-5+6 using long division The Attempt at a Solution ƒ(2)=2(2)^3-3(2)^2-5(2)+6 ƒ(2)=16-12-10+6 ƒ(2)=0 Now using long division divide 2x^3-3x^2-5x+6 by (x-2)...

Is the fundamental theorem of algebra (for polynomials on the complex plane) equivalent to the statement that any polynomial p of degree n>0 can be written p(z) = c(z - a_1 ) (z- a_2) \cdot \cdot \cdot (z - a_n ) or am I missing some subtle distinction? And if not equivalent, does the theorem...

The problem is (x+y)^2+2(X+Y)+1 and the answer is supposed to be (X+1+Y)^2 I looked it up on this calculator but the second step makes no sense. First I do not know how the +2 disappears and where the extra (X+Y) come from. Also it's explanation when you click on the black box doesn't make...

Homework Statement Hello! One of the easiest rules (when possible to apply) to factor a quadratic is to find both x-s by x1 + x2 = b x1 * x2 = c Homework Equations Please, take a look at what is written in the book. I can't grasp why x1 = -2 and x2 = 3, and not, as I thought, x1 = -3 and x2...

Homework Statement If I have the characteristic equation: -λ3 + 3λ2 + 9λ + 5 And I'm told that one of its eigenvalues is -1. How do I find the rest of the eigenvalues? Homework Equations -λ3 + 3λ2 + 9λ + 5 The Attempt at a Solution The furthest I can get is: -λ3 + 3λ2 + 9λ + 5 = (λ + 1) x...

Homework Statement (4a/a+4)+(a+2/2a) Homework Equations Just combine and then factor out The Attempt at a Solution It's actually fairly simple, but I'm having difficulty at the end. /multiply each term by opposite denominator 4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4) /combine 4a(2a)+(a+2)(a+4) /...

Homework Statement Let ## a,b,c \in \mathbb{R}^{+} ##. Prove that $$ \sqrt[3]{abc} \leq \frac{a+b+c}{3}. $$ Note: ## a,b,c ## can be expressed as ## a = r^3, b = s^3, c = t^3 ## for ## r,s,t > 0##. Homework Equations ## P(a,b,c): a,b,c \in \mathbb{R}^{+} ## ## Q(a,b,c): \sqrt[3]{abc} \leq...

Homework Statement Find the roots of z^4+4=0 and use that to factor the expression into quadratic factors with real coefficients. Homework Equations DeMoivre's formula. The Attempt at a Solution I have been able to identify they are \pm 1 \pm i but i have no idea how to factor the...

Homework Statement Determine the maximum value of the function f(x) = 4e^(-x) - e^(x) Homework Equations dasfgd The Attempt at a Solution 1. I first found the derivative of the function and it turned out to be f`(x) = -4e^-x - e^x 2. I believe that to find the maximum value of this function...

Homework Statement Factor out the polynomial and find its solutions x^3-5x^2+7x-12[/B]Homework EquationsThe Attempt at a Solution I tried to factor it, but I'm stuck in this step x^2(x-5)+7(x-5)+23= 0. I graphed the equation, and I know there is two imaginary solutions and one real positive...

I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.

(x2)3(2x3)3/(4x)2 I got 1/2x^13, is this correct?

[FONT=Lucida Grande]The number of ways in which the number $27720$ can be split into two factors which are co prime

Homework Statement If f(x) = mx^3 + gx^2 - x + 3 is divided by x - 1, the remainder is 3. If f(x) is divided by x + 3, the remainder is -1. What are the values of m and g? Homework Equations Remainder Theorum/Factor Theorum The Attempt at a Solution Here is what I did...

I wanted to factor xy+x-2y-2 I got x(y+1)-2(y-1) and got stuck I tried somethings out and noticed (x-2)(y+1)=xy+x-2y-2 so how come I did get stuck? Did I extract the 2 incorrectly with the sign? For example should I have interpreted it at -2y-2 not 2y-2?

I only see positive factors in my book. Example: 2 and 2 are factors of 4. But aren't -2 and -2? Is there some unspoken rule in math that only positive factors count? Thanks!

How do you factor $$y^4+y^2+1$$? I know that if you regroup them like this: $$(y^4+2y^2+1)-y^2$$, factoring becomes apparent: $$(y^2 + 1 + y) (y^2 + 1 - y)$$ Is there any other way to factor this? It's just a pain to have to try out these "tricks" every time I try to factor something.

Homework Statement Factor the expression ac-bd+adi+bci Homework Equations The Attempt at a Solution We can factor the variable 'a' which gives: a(c+di)-bd+bci The common factor in the remaining terms is b, and if we also factor out b we get a(c+di)-b(d+ci) But this is not the...

I came across an interesting website today, which stated that you can factor sums of squares IF 2AB is a perfect square known as the Sophie Germain’s Identity. Below is the website for your reference. http://oakroadsystems.com/math/sumsqr.htm I realize that if it is a perfect square, then the...

hi all I am stuyding how to factor equations such as :- 7x^2-9X-6 The problem i have is that it takes me too long to find 2 numbers whose sum is D and the same numbers whoes product is E. Is there any way/tips/guide on how i can achieve this quickly?

Hi, I understand the tensor product of modules as a new module in which every bilinear map becomes a linear map. But now I am trying to see the Tensor product of modules from the perspective of maps factoring through, i.e., from properties that allow a commutative triangle of maps. As a...

I am trying to write \[ \frac{a^4+b^4}{a^2+b^2} \] with nothing higher than a power of two. I know \(a^2+b^2 = (a + ib)(a - ib)\) and \(a^4 + b^4 = (a^2 + ib^2)(a^2 - ib^2)\), but I am to take the numerator down in farther in hopes of some cancelling in the denominator.

trust me this is trivial... As a kid I had a teacher fond of asking if numbers were prime. Of course at the time I had no calculator and did not have many primes remembered. I did not even know the less than square root. I came up with a method that made a simple chart of smaller than the...

Hey everybody, I thought I would share an algorithm for factoring that I designed a few years ago while working on a Z80 project. I noted the same thing Fermat must have: a^{2}-b^{2}=(a+b)(a-b) After a few moments of looking at it, I realized the translation of that simple expression: If I can...

Homework Statement Simplify the expression by factoring. 3(x+1)1/2(2x-3)5/2+10(x+1)3/2(2x-3)3/2 Homework Equations The Attempt at a Solution 3(x+1)1/2(2x-3)5/2+10(x+1)3/2(2x-3)3/2 = (x + 1)1/2(2x-3)3/2[3(2x-3) + 10(x+1)] = (x + 1)1/2(2x - 3)3/2(6x - 9 + 10x + 10) = (x + 1)1/2(2x - 3)3/2(16x...

Homework Statement 9cos(2t)=9cos2(t)-4 for all the smallest four positive solutions Homework Equations The Attempt at a Solution I've factored it and pulled out 9cos(t) and made that = u so i have u2-2u-4 That factors to 3.23606 and -1.236 Next i added 9cos(t) back in...

$\displaystyle x^4+2x^3-8x^2+18x+9$ this is what i tried, $x^4+2x^3-8x^2 = (x^2+4x)(x^2-2x)$ then, $a(x^2+4x)+b(x^2-2x)=18x$ where ab=9 did i set up my solution correctly? can you tell me where I'm wrong.

I'm looking to factor this... X^3/2 - 2x^-1/2 + x^1/2 The textbook offers the solution 1-x ^4/3 No idea how to get this result. I got X ^-3/2 (x-1)^2 I would like it if someone could shed light on how to factor x^-3/2 from things like 2x ^1/2 I assume you just keep the equation...

Hello guys, I'd like to ask you how to efficiently factorize complicated polynomial like this one for example: $$\frac{128t^4+128t^3+192t^2+32t+40}{(4t^2+1)^4}$$ I've spend more than a hour trying to decrypt and decompose the polynomial, but to no avail. For simple cubic polynomial I know...

Homework Statement Was given a matrix To find the eigenvalues I set up the characteristic equation [-1-x | 7 | -5 ] [-4 | 11-x | -6 ] [-4 | 8 | -3-x] With some dirty work I got this bad boy out, which I'm having trouble factoring -x3+7x2-15x+9Homework Equations...

any hints on how to start this problem? $12x^4+19x^3-26x^2-61x-28$

how would i start factoring this $m^8-n^8-2m^6n^2+2n^6m^2$

how would i go about properly grouping terms to start my factoring? 1. $12x^2y^3-24y^3z-6x^6+30y^4-15x^4y+4x^2y+12x^4z+10y^2-8yz$ thanks!

can you guys help me factor this polynomial. $\displaystyle 2x^2-4xy+2y^2+5x-3-5y$ $6x^2-xy+23x-2y^2-6y+20$ by the way this is what i tried in prob 1 $2(x-y)^2+5(x-y)-3$ -->> I'm stuck here. and in prob 2 i have no idea where and how to start. thanks!

Homework Statement Question + attempt: Homework Equations The Attempt at a Solution Why is it that when I expand the factored form, I don't get the original equation?

One of my quadratic formula problems required me to factor so that the product of 2050s factors added to -23, that may not be exact, but I basically asked for a new problem since it was just practice, after an hour of getting close but never the right answer, and I was wondering, other than just...

I'm having trouble factoring the following binomial... can someone try to point me in the right direction please? 4y^3+4 It has been 7 years since I took algebra and I am trying to get my review done. This one just does not make sense to me right now... Thanks!

Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :) PART A 5) Express .00024 in Sci Notation My answer: 2.4 x 10^-4 6) Factor the following completely: 3x^3 - 18x^2 + 24x My Answer: (3x)(x - 4)(x - 2)

how to force factor this into the difference of two squares. $\displaystyle x^4 + 1$

Homework Statement I want to factor polynomials. However, I want to know how it is possible to factor polynomials where the linear factors that result do not have integers but rather fractions. Should we continue factoring if there are fractions, or do we have to stop? Say we have this...

Homework Statement p(x) = x^4+10x^3+26x^2+10x+1 p(x) = a(x)b(x) where a(x) and b(x) are quadratic polynomials with integer coefficients. It is given that b(1) > a(1). Find a(3) + b(2). Homework Equations p(x) = x^4+10x^3+26x^2+10x+1The Attempt at a Solution I tried to factor the given quartic...