×
Back to all chapters

Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.

# Factorization of Rational Functions

If $$P(n)=\frac{n^3+n^2-2n}{n^2-4}+\frac{n+1}{2+n-n^2}$$, what is the value of $$\lfloor P(409) \rfloor + \lfloor P(423) \rfloor$$?

Details and assumptions

The function $$\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}$$ refers to the greatest integer smaller than or equal to $$x$$. For example, $$\lfloor 2.3 \rfloor = 2$$ and $$\lfloor -5 \rfloor = -5$$.

If $$x$$ and $$y$$ are real numbers such that $x\neq y, x+y=5, \mbox{ and } xy=2,$ what is the value of $\frac{x^8-y^8}{433(x-y)}?$

If $$\frac{x^2+y^2+z^2}{xy+yz+zx}=7$$ and $$\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}=6$$, what is the value of $$\frac{(x+y+z)^3}{xyz}$$?

Let $$a$$ be a positive number such that $a^2+\frac{1}{a^2}=3.$ If the value of $a^3+\frac{1}{a^3}$ can be expressed as $$m\sqrt{n},$$ where $$n$$ is a prime number, what is $$m+n?$$

$$x, y$$ and $$z$$ are complex numbers such that

$\begin{cases} x + y + z & = 46 \\ (x-y)^2 + (y-z)^2 + (z-x) ^ 2 & = xyz.\\ \end{cases}$

What is the value of $$\frac{ x^3 + y^3 + z^3} { xyz}$$?

×