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### How to factor $9x^2-80x-9$? [closed]

How do I factor a trinomial like this? I'm having a lot of difficulty. How do I deal with the $9x^2$?
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I have seen this lemma elsewhere. Suppose $A$ is a domain, and $f \in A[X]$. Prove that $$a - b \mid f(a) - f(b)$$ I need to prove this. $$f(a) - f(b) \equiv 0 \pmod{a-b}$$ basically. Let, $a - ... 2answers 335 views ### Different version of Gauss's Lemma Let$A$be a domain with field of fractions$K$. Let$f, g \in A[X]$with$g$monic. Show that if$f/g \in K[X]$then$f/g \in A[X]$. So I tried the direct approach by just assuming$f/g$has a ... 2answers 515 views ### Does Factor Theorem fail for non-fields? I would like to solve the following: By the Factor theorem a polynomial$f ∈ R[x]$, for$R$is a field, then has a root in$R$if and only if$(x-a)$factor. Is the same statement necessarily true ... 5answers 97 views ### Find the values of$n$that make the fraction$\frac{2n^{7}+1}{3n^{3}+2}$reducible. Question : Find the values of$n\in \mathbb{N}$that make the fraction$\frac{2n^{7}+1}{3n^{3}+2}$reducible ? I don't know any ideas or hints how I solve this question ? I think we must be ... 5answers 104 views ### Find positive integers such that$2n^3 + 5|n^4 +n+1$$$2n^3 + 5 | 2n^4 +2n +2 - 2n^4 - 5n$$ $$= 2n^3+5 | 2-3n$$ $$3n-2≥2n^3 + 5$$ is this correct? Is there a more efficient way? 2answers 852 views ### Polynomial long division with mod. Trouble with fractions. For example$4x^4 + x + 1$divided by$3x + 1$is$\frac{4x^3}{3} - \frac{4x^2}{9} + \frac{4x}{27} + \frac{23}{81}$remainder$\frac{58}{81}$. Now I want to do the same division mod$9$, but I can't ... 2answers 394 views ### Why is the remainder of any polynomial divided by a 1st degree polynomial, a constant Here is a "Math is fun" quote: "When we divide by a polynomial of degree$1$(such as "$x-3$") the remainder will have degree$0$(in other words a constant, like "$4$")." I'm hoping someone could ... 1answer 366 views ### Factor theorem for multiple variables I know that similar questions have been asked often. I have had a look at many of them, and I have found the following claim over polynomials in two variables in an answer here:$\$x-(by+c) \ | \ f(...

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