1. Remainder theorem:
If a polynomial f(x) is divided by x-α, the remainder obtained is f(α)
A polynomial f(x) is divided by x-α, if f(α)=0.
If a1 /a2 =a1 /a2= a1 /a2 , then each of these ratios is equal to
a) (ka1 +lb1 +mc1) / (ka2 +lb2 +mc2)
b) (ka1x +lb1x +mc1x) / (ka2x +lb2x+mc2x)1/x
c) (a1 b1/a2b2 )1/2 = (a1 b1 c1/a2b2 c2 )1/3
For example, a/b= 3/4, then
a/b= 3/4 = (a+3)/(b+4) = (a2 +9)1/2/(b2 +16)1/2
2. Condition for resolution into linear factors of a quadratic function:
The quadratic function ax2 +2fgh+by2 +2gx +2fy + c is resolvable into linear factors iff
abc + 2fgh - af2 -bg2 -ch2 =0 i.e.