Linked Questions

1
vote
9answers
1k views

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 $9x^2$?
13
votes
2answers
930 views

Proof of the single factor theorem over an arbitrary commutative ring

Theorem (Single factor theorem) Let $R$ be a commutative ring, and let $P\in R[X]$, where $R[X]$ is the polynomial ring over the indeterminate $X$. Suppose $P(\alpha)=0$. Then $X-\alpha$ divides $P(...
6
votes
2answers
597 views

$\dfrac{\mathbb{Z}[x]}{(x^2 +5)}$ is isomorphic to $\mathbb{Z}[\sqrt{-5}]$

Let $H : \mathbb{Z}[x] \rightarrow \mathbb{Z}[\sqrt{-5}]$ be the evaluation homomorphism given by $H(f) = f(\sqrt{-5})$. We know that $H$ is surjective. I wanna to show that $\ker(H) = (x^2 +5)$. I ...
4
votes
1answer
3k views

Division algorithm for polynomials in R[x], where R is a commutative ring with unity.

Is there a division algorithm for polynomials in R[x], where R is a commutative ring with unity? All the algebra books I read mention division algorithm for polynomials in F[x], where F is a field. ...
4
votes
4answers
236 views

Factorizing a polynomial $f$ in $A[x]$ (with $A$ commutative), where $f$ has a zero in its field of fractions

Let $A$ be a commutative ring and $S$ a multiplicative subset of $A$ generated by $s\in A$ which is not a zero-divisor. Consider the polynomial ring $A[x]$. Given a polynomial $f\in A[x]$, suppose ...
13
votes
2answers
498 views

Show that $A[X]/(aX+b)$ is an integral domain

Let $A$ be an integral domain, $a$ and $b \in A-\{0\}$, and let $B = A[X]/(aX+b)$. Show that, if $Aa \cap Ab=Aab$, then $B$ is an integral domain. My attempt at proof (following a hint). Denote by $K$...
4
votes
2answers
1k views

Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring

I read "Algebra: Chapter 0" by P.Aluffi. I encountered a topic where it says you can divide any polynomial in $R[x]$($R$ is any ring) by a monic polynomial(that is, a polynomial of the form $x^d + \...
4
votes
4answers
204 views

Show that $a - b \mid f(a) - f(b)$

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 - ...
5
votes
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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
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?
1
vote
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 ...
0
votes
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 ...
3
votes
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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