a) \(({x^2}-2x + 1)\left( {x-1} \right)\)
\( = {x^2}.\left( {x - 1} \right) + \left( { - 2x} \right).\left( {x - 1} \right) \)\(\,+ 1.\left( {x - 1} \right)\)
\( = {x^2}.x + {x^2}.\left( { - 1} \right) + \left( { - 2x} \right).x \)\(+ \left( { - 2x} \right).\left( { - 1} \right) + 1.x + 1.\left( { - 1} \right)\)
\(= {x^3} - {x^{2}} - 2{x^2} + 2x + x-1\)
\( = {x^3} + \left( { - {x^2} - 2{x^2}} \right) + \left( {2x + x} \right) - 1\)
\(= {x^3} - 3{x^2} + 3x-1.\)
b) \(({x^3}-2{x^{2}} + x - 1)\left( {5-x} \right)\)
\( = {x^3}\left( {5 - x} \right) - 2{x^2}\left( {5 - x} \right) + x\left( {5 - x} \right)\)\(\, - 1.\left( {5 - x} \right)\)
\(= {x^3}.5 + {x^3}.\left( { - x} \right) + ( - 2{x^2}).5 \)\(+ ( - 2{x^2})\left( { - x} \right) + x.5 + x\left( { - x} \right) \)\(+ \left( { - 1} \right).5 + \left( { - 1} \right).\left( { - x} \right)\)
\( = 5{x^3}-{x^4}-10{x^2} + 2{x^3} + 5x-{x^2}\)\(-5 + x\)
\( = - {x^4} + \left( {5{x^3} + 2{x^3}} \right) + \left( { - 10{x^2} - {x^2}} \right)\)\(\, + \left( {5x + x} \right) - 5\)
\( = - {x^4} + 7{x^3}-11{x^2} + 6x - 5.\)
Suy ra kết quả của phép nhân:
\(\matrix{ {\left( {{x^3}-2{x^2} + x - 1} \right)\left( {x - 5} \right)} \hfill \cr { = \left( {{x^3}-2{x^2} + x - 1} \right)\left[ { - \left( {5 - x} \right)} \right]} \hfill \cr { = - \left( {{x^3}-2{x^2} + x - 1} \right)\left( {5-x} \right)} \hfill \cr { = - \left( { - {x^4} + 7{x^3}-11{x^2} + 6x - 5} \right)} \hfill \cr { = {x^4} - 7{x^3} + 11{x^2} - 6x + 5} \hfill \cr} \)
(Ở đây ta biến đổi: \(x - 5 = - (5 - x)\))
