Skip to main content

[Community Question] Algebra-Precalculus: Find the remainder of the division of $x^n+5$ with $x^3+10x^2+25x$

One of our user asked:

Find the remainder of the division of $x^n+5$ with $x^3+10x^2+25x$ over $\mathbb{Q}$

What I tried to do is to write $x^n+5=p(x)(x^3+10x^2+25x)+Ax^2+Bx+C$, where $p(x)$ is a polynomial of degree $n-3$. If I set $x=0$ I obtain that $C=0$.

Now $x^3+10x^2+25x=x(x+5)^2$ and by setting $x=-5$, I get that

$5A-B=(-1)^n5^{n-1}$

But I need one more equation to be able to find the coefficients of the remainder, and I can't get one. What should I do?


Labels:

Comments

Post a Comment

Popular posts from this blog

[Community Question] Linear-algebra: non-negative matrix satisfying two conditions

One of our user asked: A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$ . I want to find a non-negative matrix $B$ satisfying the following two conditions: (1) $(I-B)^{-1}$ exists but not non-negative. Here $I$ is the identity matrix. (2) There is a non-zero and non-negative vector $\vec{d}$ such that $(I-B)^{-1}\vec{d}\ge 0$ . I tried all the $2\times 2$ matrices, but it did not work. I conjecture that such a $B$ does not exist, but don't know how to prove it.
Post a Comment
Read more

Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$ , I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$ . i try several ways but I have not been successful, some help ?? thank you
Post a Comment
Read more