Advanced Factorization

Factorization of Rational Functions


If P(n)=n3+n22nn24+n+12+nn2P(n)=\frac{n^3+n^2-2n}{n^2-4}+\frac{n+1}{2+n-n^2}, what is the value of P(409)+P(423)\lfloor P(409) \rfloor + \lfloor P(423) \rfloor?

Details and assumptions

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

If xx and yy are real numbers such that xy,x+y=5, and xy=2,x\neq y, x+y=5, \mbox{ and } xy=2, what is the value of x8y8433(xy)?\frac{x^8-y^8}{433(x-y)}?

If x2+y2+z2xy+yz+zx=7\frac{x^2+y^2+z^2}{xy+yz+zx}=7 and y+zx+z+xy+x+yz=6\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}=6, what is the value of (x+y+z)3xyz\frac{(x+y+z)^3}{xyz}?

Let aa be a positive number such that a2+1a2=3.a^2+\frac{1}{a^2}=3. If the value of a3+1a3a^3+\frac{1}{a^3} can be expressed as mn,m\sqrt{n}, where nn is a prime number, what is m+n?m+n?

x,yx, y and zz are complex numbers such that

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

What is the value of x3+y3+z3xyz \frac{ x^3 + y^3 + z^3} { xyz} ?


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