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 a _{1} /a_{2} =a_{1} /a_{2= }a_{1} /a_{2} , then each of these ratios is equal toa) (ka _{1} +lb_{1} +mc_{1)} / (ka_{2} +lb_{2} +mc_{2)b) }(ka_{1}^{x} +lb_{1}^{x} +mc_{1}^{x}_{)} / (ka_{2}^{x} +lb_{2}^{x}+mc_{2}^{x}_{)}^{1/x} c) (a _{1} b_{1}/a_{2}b_{2} )^{1/2} = (a_{1} b_{1} c_{1}/a_{2}b_{2 }c_{2} )^{1/3} For example, a/b= 3/4, thena/b= 3/4 = (a+3)/(b+4) = (a ^{2} +9)^{1/2}/(b^{2} +16)^{1/2} 2. Condition for resolution into linear factors of a quadratic function:The quadratic function ax ^{2} +2fgh+by^{2} +2gx +2fy + c is resolvable into linear factors iffabc + 2fgh - af ^{2} -bg^{2} -ch^{2} =0 i.e. |