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15. some important results

IN THIS CHAPTER:



1. Remainder theorem:
    If a polynomial f(x) is divided by x-α, the remainder obtained is f(α)

   Factor theorem:
   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.



 
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