Đáp án:

\(\begin{array}{l}
1.6\\
a,\\
{x^5} + {y^5}\\
b,\\
{x^3} + {x^2}.\left( {a + b + c} \right) + x.\left( {ab + bc + ca} \right) + abc\\
c,\\
{A^k} - {B^k}\\
1.7\\
a,\\
{x^3} + {y^3} + {z^3} = \left( {x + y + z} \right).\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right) + 3xyz\\
b,\\
\left( {{x^2} + x + 1} \right)\left( {{x^5} - {x^4} + {x^3} - x + 1} \right) = {x^7} + {x^5} + 1\\
8,\\
a,\\
P\left( x \right) = 94\\
b,\\
Q\left( x \right) = 1\\
1.9\\
a,\\
f\left( x \right) = 3\\
b,\\
h\left( x \right) = 2
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
1.6\\
a,\\
\left( {x + y} \right)\left( {{x^4} - {x^3}y + {x^2}{y^2} - x{y^3} + {y^4}} \right)\\
 = x.\left( {{x^4} - {x^3}y + {x^2}{y^2} - x{y^3} + {y^4}} \right) + y.\left( {{x^4} - {x^3}y + {x^2}{y^2} - x{y^3} + {y^4}} \right)\\
 = {x^5} - {x^4}y + {x^3}{y^2} - {x^2}{y^3} + x{y^4} + {x^4}y - {x^3}{y^2} + {x^2}{y^3} - x{y^4} + {y^5}\\
 = {x^5} + \left( { - {x^4}y + {x^4}y} \right) + \left( {{x^3}{y^2} - {x^3}{y^2}} \right) + \left( { - {x^2}{y^3} + {x^2}{y^3}} \right) + \left( {x{y^4} - x{y^4}} \right) + {y^5}\\
 = {x^5} + {y^5}\\
b,\\
\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right)\\
 = \left( {{x^2} + xb + ax + ab} \right)\left( {x + c} \right)\\
 = x.\left( {{x^2} + xb + xa + ab} \right) + c.\left( {{x^2} + xb + xa + ab} \right)\\
 = {x^3} + {x^2}b + {x^2}a + x.ab + c{x^2} + bcx + cax + abc\\
 = {x^3} + \left( {{x^2}b + {x^2}a + {x^2}c} \right) + \left( {x.ab + bcx + cax} \right) + abc\\
 = {x^3} + {x^2}.\left( {a + b + c} \right) + x.\left( {ab + bc + ca} \right) + abc\\
c,\\
\left( {A - B} \right).\left( {{A^{k - 1}} + {A^{k - 2}}B + {A^{k - 3}}{B^2} + .... + A{B^{k - 2}} + {B^{k - 1}}} \right)\\
 = A.\left( {{A^{k - 1}} + {A^{k - 2}}B + {A^{k - 3}}{B^2} + .... + A{B^{k - 2}} + {B^{k - 1}}} \right) - B.\left( {{A^{k - 1}} + {A^{k - 2}}B + {A^{k - 3}}{B^2} + .... + A{B^{k - 2}} + {B^{k - 1}}} \right)\\
 = {A^k} + {A^{k - 1}}B + {A^{k - 2}}{B^2} + .... + {A^2}{B^{k - 2}} + A.{B^{k - 1}} - \left( {{A^{k - 1}}B + {A^{k - 2}}{B^2} + {A^{k - 3}}{B^3} + .... + A.{B^{k - 1}} + {B^k}} \right)\\
 = {A^k} + \left( {{A^{k - 1}}B + {A^{k - 2}}{B^2} + .... + {A^2}{B^{k - 2}} + A.{B^{k - 1}}} \right) - \left( {{A^{k - 1}}B + {A^{k - 2}}{B^2} + .... + {A^2}{B^{k - 2}} + A.{B^{k - 1}}} \right) - {B^k}\\
 = {A^k} - {B^k}\\
1.7\\
a,\\
{x^3} + {y^3} + {z^3}\\
 = \left( {{x^3} + 3{x^2}y + 3x{y^2} + {y^3}} \right) + {z^3} - \left( {3{x^2}y + 3x{y^2}} \right)\\
 = {\left( {x + y} \right)^3} + {z^3} - \left( {3{x^2}y + 3x{y^2} + 3xyz} \right) + 3xyz\\
 = \left[ {\left( {x + y} \right) + z} \right].\left[ {{{\left( {x + y} \right)}^2} - \left( {x + y} \right).z + {z^2}} \right] - 3xy.\left( {x + y + z} \right) + 3xyz\\
 = \left( {x + y + z} \right).\left( {{x^2} + 2xy + {y^2} - xz - yz + {z^2}} \right) - 3xy\left( {x + y + z} \right) + 3xyz\\
 = \left( {x + y + z} \right).\left[ {\left( {{x^2} + 2xy + {y^2} - xz - yz + {z^2}} \right) - 3xy} \right] + 3xyz\\
 = \left( {x + y + z} \right).\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right) + 3xyz\\
b,\\
\left( {{x^2} + x + 1} \right)\left( {{x^5} - {x^4} + {x^3} - x + 1} \right)\\
 = \left( {{x^2} + x + 1} \right).\left[ {\left( {{x^5} - {x^4}} \right) + \left( {{x^3} - {x^2}} \right) + \left( {{x^2} - x} \right) + 1} \right]\\
 = \left( {{x^2} + x + 1} \right).\left[ {{x^4}.\left( {x - 1} \right) + {x^2}.\left( {x - 1} \right) + x.\left( {x - 1} \right) + 1} \right]\\
 = \left( {{x^2} + x + 1} \right).\left[ {\left( {x - 1} \right).\left( {{x^4} + {x^2} + x} \right) + 1} \right]\\
 = \left( {{x^2} + x + 1} \right).\left( {x - 1} \right).\left( {{x^4} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^2} + x.1 + {1^2}} \right).\left( {x - 1} \right).\left( {{x^4} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^3} - {1^3}} \right).\left( {{x^4} + {x^2} + x} \right) + {x^2} + x + 1\\
 = \left( {{x^3} - 1} \right)\left( {{x^4} + {x^2} + x} \right) + {x^2} + x + 1\\
 = {x^3}.\left( {{x^4} + {x^2} + x} \right) - 1.\left( {{x^4} + {x^2} + x} \right) + {x^2} + x + 1\\
 = {x^7} + {x^5} + {x^4} - {x^4} - {x^2} - x + {x^2} + x + 1\\
 = {x^7} + {x^5} + 1\\
8,\\
a,\\
P\left( x \right) = {x^7} - 80{x^6} + 80{x^5} - 80{x^4} + 80{x^3} - 80{x^2} + 80x + 15\\
 = \left( {{x^7} - 79{x^6}} \right) + \left( { - {x^6} + 79{x^5}} \right) + \left( {{x^5} - 79{x^4}} \right) + \left( { - {x^4} + 79{x^3}} \right) + \left( {{x^3} - 79{x^2}} \right) + \left( { - {x^2} + 79x} \right) + \left( {x - 79} \right) + 94\\
 = {x^6}.\left( {x - 79} \right) - {x^5}.\left( {x - 79} \right) + {x^4}.\left( {x - 79} \right) - {x^3}.\left( {x - 79} \right) + {x^2}.\left( {x - 79} \right) - x.\left( {x - 79} \right) + \left( {x - 79} \right) + 94\\
 = \left( {x - 79} \right).\left( {{x^6} - {x^5} + {x^4} - {x^3} + {x^2} - x + 1} \right) + 94\\
x = 79 \Rightarrow x - 79 = 0 \Rightarrow P\left( x \right) = 94\\
b,\\
Q\left( x \right) = {x^{10}} - 15{x^9} + 15{x^8} - 15{x^7} + .... + 15{x^2} - 15x + 15\\
 = \left( {{x^{10}} - 14{x^9}} \right) + \left( { - {x^9} + 14{x^8}} \right) + \left( {{x^8} - 14{x^7}} \right) + ..... + \left( { - {x^3} + 14{x^2}} \right) + \left( {{x^2} - 14x} \right) + \left( { - x + 14} \right) + 1\\
 = {x^9}.\left( {x - 14} \right) - {x^8}.\left( {x - 14} \right) + {x^7}.\left( {x - 14} \right) - ..... - {x^2}.\left( {x - 14} \right) + x.\left( {x - 14} \right) - \left( {x - 14} \right) + 1\\
 = \left( {x - 14} \right).\left( {{x^9} - {x^8} + {x^7} - ..... - {x^2} + x - 1} \right) + 1\\
x = 14 \Rightarrow x - 14 = 0 \Rightarrow Q\left( x \right) = 1\\
1.9\\
a,\\
f\left( x \right) = x\left( {2x + 1} \right) - {x^2}.\left( {x + 2} \right) + {x^3} - x + 3\\
 = \left( {2{x^2} + x} \right) - \left( {{x^3} + 2{x^2}} \right) + {x^3} - x + 3\\
 = 2{x^2} + x - {x^3} - 2{x^2} + {x^3} - x + 3\\
 = \left( { - {x^3} + {x^3}} \right) + \left( {2{x^2} - 2{x^2}} \right) + \left( {x - x} \right) + 3\\
 = 3\\
b,\\
h\left( x \right) = \left( {x + 1} \right)\left( {{x^2} - x + 1} \right) - \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\\
 = \left( {x + 1} \right).\left( {{x^2} - x.1 + {1^2}} \right) - \left( {x - 1} \right).\left( {{x^2} + x.1 + {1^2}} \right)\\
 = \left( {{x^3} + {1^3}} \right) - \left( {{x^3} - {1^3}} \right)\\
 = {x^3} + 1 - {x^3} + 1\\
 = 2
\end{array}\)