Đáp án:

\(\begin{array}{l}
1,\\
a,\\
{x^2} + 4xy + 4{y^2}\\
b,\\
9{x^2} - 12xy + 4{y^2}\\
c,\\
4{x^2} - 2x + \dfrac{1}{4}\\
d,\\
\dfrac{{{x^2}}}{4} - {y^2}\\
e,\\
{x^3} + {x^2} + \dfrac{1}{3}x + \dfrac{1}{{27}}\\
f,\\
{x^3} - 8\\
2,\\
a,\\
{\left( {x + y} \right)^2} - {y^2} = x.\left( {x + 2y} \right)\\
b,\\
{\left( {{x^2} + {y^2}} \right)^2} - {\left( {2xy} \right)^2} = {\left( {x - y} \right)^2}.{\left( {x + y} \right)^2}\\
c,\\
{\left( {x + y} \right)^3} = x.{\left( {x - 3y} \right)^2} + y.{\left( {y - 3x} \right)^2}\\
3,\\
a,\\
{\left( {a + b} \right)^3} + {\left( {a - b} \right)^3} = 2a.\left( {{a^2} + 3{b^2}} \right)\\
b,\\
{\left( {a + b} \right)^3} - {\left( {a - b} \right)^3} = 2b.\left( {3{a^2} + {b^2}} \right)\\
4,\\
a,\\
\left( {x + 2y} \right).\left( {{x^2} - 2xy + 4{y^2}} \right)\\
b,\\
\left( {{a^2} - b} \right)\left( {{a^4} + {a^2}b + {b^2}} \right)\\
c,\\
\left( {2y - 5} \right)\left( {4{y^2} + 10y + 25} \right)\\
d,\\
\left( {2z + 3} \right).\left( {4{z^2} - 6z + 9} \right)\\
5,\\
a,\\
{1001^2} = 1002001\\
29,9.30,1 = 899,99\\
b,\,\,\,\,100
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
1,\\
a,\\
{\left( {x + 2y} \right)^2} = {x^2} + 2.x.2y + {\left( {2y} \right)^2} = {x^2} + 4xy + 4{y^2}\\
b,\\
{\left( {3x - 2y} \right)^2} = {\left( {3x} \right)^2} - 2.3x.2y + {\left( {2y} \right)^2} = 9{x^2} - 12xy + 4{y^2}\\
c,\\
{\left( {2x - \dfrac{1}{2}} \right)^2} = {\left( {2x} \right)^2} - 2.2x.\dfrac{1}{2} + {\left( {\dfrac{1}{2}} \right)^2} = 4{x^2} - 2x + \dfrac{1}{4}\\
d,\\
\left( {\dfrac{x}{2} - y} \right).\left( {\dfrac{x}{2} + y} \right) = {\left( {\dfrac{x}{2}} \right)^2} - {y^2} = \dfrac{{{x^2}}}{4} - {y^2}\\
e,\\
{\left( {x + \dfrac{1}{3}} \right)^3} = {x^3} + 3.{x^2}.\dfrac{1}{3} + 3.x.{\left( {\dfrac{1}{3}} \right)^2} + {\left( {\dfrac{1}{3}} \right)^3}\\
 = {x^3} + {x^2} + 3.x.\dfrac{1}{9} + \dfrac{1}{{27}}\\
 = {x^3} + {x^2} + \dfrac{1}{3}x + \dfrac{1}{{27}}\\
f,\\
\left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right)\\
 = \left( {x - 2} \right).\left( {{x^2} + x.2 + {2^2}} \right)\\
 = {x^3} - {2^3} = {x^3} - 8\\
2,\\
a,\\
{\left( {x + y} \right)^2} - {y^2}\\
 = \left[ {\left( {x + y} \right) - y} \right].\left[ {\left( {x + y} \right) + y} \right]\\
 = \left( {x + y - y} \right).\left( {x + y + y} \right)\\
 = x.\left( {x + 2y} \right)\\
b,\\
{\left( {{x^2} + {y^2}} \right)^2} - {\left( {2xy} \right)^2}\\
 = \left[ {\left( {{x^2} + {y^2}} \right) - 2xy} \right].\left[ {\left( {{x^2} + {y^2}} \right) + 2xy} \right]\\
 = \left( {{x^2} - 2xy + {y^2}} \right).\left( {{x^2} + 2xy + {y^2}} \right)\\
 = {\left( {x - y} \right)^2}.{\left( {x + y} \right)^2}\\
c,\\
{\left( {x + y} \right)^3}\\
 = {x^3} + 3{x^2}y + 3x{y^2} + {y^3}\\
 = {x^3} - 6{x^2}y + 9{x^2}y - 6x{y^2} + 9x{y^2} + {y^3}\\
 = \left( {{x^3} - 6{x^2}y + 9x{y^2}} \right) + \left( {{y^3} - 6{x^2}y + 9{x^2}y} \right)\\
 = x.\left( {{x^2} - 6xy + 9{y^2}} \right) + y.\left( {{y^2} - 6xy + 9{x^2}} \right)\\
 = x.\left[ {{x^2} - 2.x.3y + {{\left( {3y} \right)}^2}} \right] + y.\left[ {{y^2} - 2.y.3x + {{\left( {3x} \right)}^2}} \right]\\
 = x.{\left( {x - 3y} \right)^2} + y.{\left( {y - 3x} \right)^2}\\
3,\\
a,\\
{\left( {a + b} \right)^3} + {\left( {a - b} \right)^3}\\
 = \left[ {\left( {a + b} \right) + \left( {a - b} \right)} \right].\left[ {{{\left( {a + b} \right)}^2} - \left( {a + b} \right).\left( {a - b} \right) + {{\left( {a - b} \right)}^2}} \right]\\
 = \left( {a + b + a - b} \right).\left[ {\left( {{a^2} + 2ab + {b^2}} \right) - \left( {{a^2} - {b^2}} \right) + \left( {{a^2} - 2ab + {b^2}} \right)} \right]\\
 = 2a.\left( {{a^2} + 2ab + {b^2} - {a^2} + {b^2} + {a^2} - 2ab + {b^2}} \right)\\
 = 2a.\left( {{a^2} + 3{b^2}} \right)\\
b,\\
{\left( {a + b} \right)^3} - {\left( {a - b} \right)^3}\\
 = \left[ {\left( {a + b} \right) - \left( {a - b} \right)} \right].\left[ {{{\left( {a + b} \right)}^2} + \left( {a + b} \right).\left( {a - b} \right) + {{\left( {a - b} \right)}^2}} \right]\\
 = \left( {a + b - a + b} \right).\left[ {\left( {{a^2} + 2ab + {b^2}} \right) + \left( {{a^2} - {b^2}} \right) + \left( {{a^2} - 2ab + {b^2}} \right)} \right]\\
 = 2b.\left( {{a^2} + 2ab + {b^2} + {a^2} - {b^2} + {a^2} - 2ab + {b^2}} \right)\\
 = 2b.\left( {3{a^2} + {b^2}} \right)\\
4,\\
a,\\
{x^3} + 8{y^3}\\
 = {x^3} + {\left( {2y} \right)^3}\\
 = \left( {x + 2y} \right).\left[ {{x^2} - x.2y + {{\left( {2y} \right)}^2}} \right]\\
 = \left( {x + 2y} \right).\left( {{x^2} - 2xy + 4{y^2}} \right)\\
b,\\
{a^6} - {b^3}\\
 = {\left( {{a^2}} \right)^3} - {b^3}\\
 = \left( {{a^2} - b} \right).\left[ {{{\left( {{a^2}} \right)}^2} + {a^2}.b + {b^2}} \right]\\
 = \left( {{a^2} - b} \right)\left( {{a^4} + {a^2}b + {b^2}} \right)\\
c,\\
8{y^3} - 125\\
 = {\left( {2y} \right)^3} - {5^3}\\
 = \left( {2y - 5} \right)\left[ {{{\left( {2y} \right)}^2} + 2y.5 + {5^2}} \right]\\
 = \left( {2y - 5} \right)\left( {4{y^2} + 10y + 25} \right)\\
d,\\
8{z^3} + 27\\
 = {\left( {2z} \right)^3} + {3^3}\\
 = \left( {2z + 3} \right).\left[ {{{\left( {2z} \right)}^2} - 2z.3 + {3^2}} \right]\\
 = \left( {2z + 3} \right).\left( {4{z^2} - 6z + 9} \right)\\
5,\\
a,\\
{1001^2} = {\left( {1000 + 1} \right)^2}\\
 = {1000^2} + 2.1000.1 + {1^2}\\
 = 1000000 + 2000 + 1\\
 = 1002001\\
29,9.30,1\\
 = \left( {30 - 0,1} \right).\left( {30 + 0,1} \right)\\
 = {30^2} - {0,1^2}\\
 = 900 - 0,01\\
 = 899,99\\
b,\\
{\left( {31,8} \right)^2} - 2.31,8.21,8 + {\left( {21,8} \right)^2}\\
 = {\left( {31,8 - 21,8} \right)^2}\\
 = {10^2} = 100
\end{array}\)