"Factoring"
The word factoring is in quotes, because these substitutions are a bit more
off-the-wall than just factoring. You'll see:
1+x4 = (x2-sqrt(2)x+1)(x2+sqrt(2)x+1)
The Quaternion Identity
So called because if u=a-bi-cj-dk and U=A+Bi+Cj+Dk, |uU| = |u||U|.
Squaring both sides gives:
(a2+b2+c2+d2)(A2+B2+C2+D2)
= (aA+bB+cC+dD)2+(aB-bA+cD-dC)2+(aC-cA+dB-bD)2+(aD-dA+bC-cB)2
Thus, the product of any two numbers, each of which can be expressed as the
sum of four squares, can also be expressed as the sum of four squares.
The Complex Product Identity
The product of any two numbers, each of which can be expressed as the sum
of two squares, can also be expressed as the sum of two squares:
(a2+b2)(A2+B2) = (aA+bB)2+(aB-bA)2
A variant of the Complex Product Identity shows that the product of two
numbers, each of which has the form a2+3b2 can also be
expressed in the form a2+3b2.
(a2+3b2)(A2+3B2)=(aA-3bB)2+3(aB+bA)2
Internet References
Related pages on this website
Integral of
sqrt(tan(x)) -- 1+x4 = (x2-sqrt(2)x+1)(x2+sqrt(2)x+1)
Sums of Squares -- proof that any
number can be expressed as the sum of four squares, using the Quaternion and
Complex Product identities.
Trig Equivalences and Table
of Integrals give examples of many more clever substitutions.
