Fun with factorization – Share your favorite!

 

Here are some examples.

 

x^3 + y^3 + z^3 - 3 \, x \,y \,z \; = \; (x + y + z) \,(x^2 + y^2 + z^2 - x \,y - y \,z - z \,x)
x^3 - y^3 - z^3 - 3 \, x \, y \, z \; = \; (x-y-z) \,(x^2 + x \, y + x \, z + y^2 - y \, z + z^2)
 
x^3 + 3 \, x \, y \, z + y^3 + z^3 - (x^2 \, y + x \, y^2 + x \, z^2 + x^2 \, z + y^2 \, z + y \, z^2)
x \, (x-y) \, (x-z) + y \, (y-x) \, (y-z) + z \, (z-x) \, (z-y)

 

(x + y + z)^4 - (y + z)^4 - (z + x)^4 - (x + y)^4 + x^4 + y^4 + z^4 \; = \; 12 \, x \, y \, z \, (x+y+z)
 

x^4 + y^4 - z^4 - 2 \, x^2 \, y^2 + 4 \, x \, y \, z^2
= \; (x + y - z) \,(x + y + z) \,(x^2 - 2 \, x \, y + y ^2 + z^2)
 

x^2 \,(y^3 - z^3) + y^2 \,(z^3 - x^3) + z^2 \,(x^3 - y^3)
= \; (x - y) \, (x-z) \, (z - y) \, (x \, y + x \, z + y \, z)
 

(yz + zx + xy)^3 - y^3 \, z^3 - z^3 \, x^3 - x^3 \, y^3
= \; 3 \, x \, y \, z \, (x+y) \, (x+z) \, (y+z)
 

x^3 \, y^3 + y^3 \, z^3 + z^3 \, x^3 - x^4 \, y \, z - x \, y^4 \, z - x \, y \, z^4
= \; (x \, z - y^2) \, (y \, z - x^2) \, (x \, y - z^2)
 

2(x^4 + y^4 + z^4 + n^4) - (x^2 + y^2 + z^2 + n^2)^2 + 8 \, x \, y \, z \, n
= (n+x-y-z) \, (n-x+y-z) \, (n-x-y+z) \, (n+x+y+z)
 

6 \,(x^5 + y^5 + z^5) - 5 \,(x^2 + y^2 + z^2) \,(x^3 + y^3 + z^3)
= \; (x+y+z)^2 \, (x^3 - 2 \, x^2 \, y - 2 \, x^2 \, z - 2 \, x \, y^2 + 6 \, x \, y \, z - 2 \, x \, z^2 + y^3 - 2 \, y^2 \, z - 2 \, y \, z^2 + z^3)

 
2 \,(a^2 + b^2 + c^2 - a \, b - b \, c - c \, a) \; = \; (a-b)^2 + (b-c)^2 + (c-a)^2

 
x^4+y^4+z^4 + x \, y \, z \,(x+y+z) - 2 \,(x^2 \, y^2 + y^2 \, z^2 + z^2 \, x^2)
= (x+y+z) \, (x^3 - x^2 \, y - x^2 \, z - x \, y^2 + 3 \, x \, y \, z - x \, z^2 + y^3 - y^2 \, z - y \, z^2 + z^3)

 
(x^3+y^3+z^3)^2 + 3(x \, y \, z)^2 - 4 \,(y^3 \, z^3 + z^3 \, x^3 + x^3 \, y^3)
= \; A \cdot B

A = \; (x^2 + x \, y + x \, z + y^2 + y \, z+ z^2)
B = \; (x^4-x^3 \, y-x^3 \, z+x^2 \, y \, z-x \, y^3+x \, y^2 \, z+x \, y \, z^2-x \, z^3+y^4-y^3 \, z-y \, z^3+z^4)

(x^3+y^3+z^3)^2 + 3 \, (x \, y \, z)^2 - 4 \, (y^3 \, z^3 + z^3 \, x^3 + x^3 \, y^3)
= \; x^6 + (y^3 - z^3)^2 + 3 (x \, y \, z)^2 - 2 x^3 \, z^3 - 2 x^3 \, y^3

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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