Đáp án:
\(\begin{array}{l}
E = 16\\
F = \dfrac{5}{4}\left( {{x^2} + {y^2} + {z^2}} \right) - \left( {xy + yz + zx} \right)\\
G = - 65\\
H = 2{x^4} + 12{x^2} + 2
\end{array}\)
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
*)\\
E = {\left( {2x - 1} \right)^2} - 2\left( {2x - 1} \right)\left( {2x + 3} \right) + {\left( {2x + 3} \right)^2}\\
= {\left( {2x - 1} \right)^2} - 2.\left( {2x - 1} \right).\left( {2x + 3} \right) + {\left( {2x + 3} \right)^2}\\
= {\left[ {\left( {2x - 1} \right) - \left( {2x + 3} \right)} \right]^2}\\
= {\left( {2x - 1 - 2x - 3} \right)^2}\\
= {\left( { - 4} \right)^2}\\
= 16\\
*)\\
F = {\left( {x - \dfrac{1}{2}y} \right)^2} + {\left( {y - \dfrac{1}{2}z} \right)^2} + {\left( {z - \dfrac{1}{2}x} \right)^2}\\
= \left[ {{x^2} - 2.x.\dfrac{1}{2}y + {{\left( {\dfrac{1}{2}y} \right)}^2}} \right] + \left[ {{y^2} - 2.y.\dfrac{1}{2}z + {{\left( {\dfrac{1}{2}z} \right)}^2}} \right] + \left[ {{z^2} - 2.z.\dfrac{1}{2}x + {{\left( {\dfrac{1}{2}x} \right)}^2}} \right]\\
= \left( {{x^2} - xy + \dfrac{1}{4}{y^2}} \right) + \left( {{y^2} - yz + \dfrac{1}{4}{z^2}} \right) + \left( {{z^2} - zx + \dfrac{1}{4}{x^2}} \right)\\
= \left( {{x^2} + \dfrac{1}{4}{x^2}} \right) + \left( {\dfrac{1}{4}{y^2} + {y^2}} \right) + \left( {\dfrac{1}{4}{z^2} + {z^2}} \right) + \left( { - xy - yz - zx} \right)\\
= \dfrac{5}{4}{x^2} + \dfrac{5}{4}{y^2} + \dfrac{5}{4}{z^2} - \left( {xy + yz + zx} \right)\\
= \dfrac{5}{4}\left( {{x^2} + {y^2} + {z^2}} \right) - \left( {xy + yz + zx} \right)\\
*)\\
G = \left( {x - 3} \right)\left( {x + 3} \right)\left( {{x^2} + 9} \right) - \left( {{x^2} - 4} \right)\left( {{x^2} + 4} \right)\\
= \left( {{x^2} - {3^2}} \right).\left( {{x^2} + 9} \right) - \left[ {{{\left( {{x^2}} \right)}^2} - {4^2}} \right]\\
= \left( {{x^2} - 9} \right).\left( {{x^2} + 9} \right) - \left( {{x^4} - 16} \right)\\
= \left[ {{{\left( {{x^2}} \right)}^2} - {9^2}} \right] - \left( {{x^4} - 16} \right)\\
= \left( {{x^4} - 81} \right) - \left( {{x^4} - 16} \right)\\
= {x^4} - 81 - {x^4} + 16\\
= - 65\\
H = {\left( {x + 1} \right)^4} + {\left( {x - 1} \right)^4}\\
= \left[ {{{\left( {x + 1} \right)}^4} - 2.{{\left( {x + 1} \right)}^2}.{{\left( {x - 1} \right)}^2} + {{\left( {x - 1} \right)}^4}} \right] + 2.{\left( {x + 1} \right)^2}.{\left( {x - 1} \right)^2}\\
= {\left[ {{{\left( {x + 1} \right)}^2} - {{\left( {x - 1} \right)}^2}} \right]^2} + 2.{\left[ {\left( {x - 1} \right)\left( {x + 1} \right)} \right]^2}\\
= {\left[ {\left( {{x^2} + 2x + 1} \right) - \left( {{x^2} - 2x + 1} \right)} \right]^2} + 2.{\left[ {{x^2} - 1} \right]^2}\\
= {\left( {{x^2} + 2x + 1 - {x^2} + 2x - 1} \right)^2} + 2.\left[ {{{\left( {{x^2}} \right)}^2} - 2.{x^2}.1 + {1^2}} \right]\\
= {\left( {4x} \right)^2} + 2.\left( {{x^4} - 2{x^2} + 1} \right)\\
= 16{x^2} + 2{x^4} - 4{x^2} + 2\\
= 2{x^4} + 12{x^2} + 2
\end{array}\)
