Đáp án:
\(\begin{array}{l}
4,\\
\left( {x - y} \right)\left( {2x - 3y} \right)\\
5,\\
\left( {x + y} \right)\left( {y - 1} \right)\\
6,\\
\left( {5 - x + 2y} \right)\left( {5 + x - 2y} \right)\\
7,\\
\left( {y + z} \right).\left( {x - 2} \right)\\
8,\\
\left( {5{x^2} - 9{y^2}} \right)\left( {5{x^2} + 9{y^2}} \right)\\
9,\\
\left( {x - 3y - 5z} \right)\left( {x - 3y + 5z} \right)\\
10,\\
3.\left( {x - y} \right)\left( {x + y - 4} \right)\\
11,\\
4x.\left( {x + y - 4} \right)\left( {x + y + 4} \right)\\
12,\\
\left( {x - 1} \right)\left( {x - 7} \right)\\
13,\\
\left( {x + 4} \right)\left( {x + 5} \right)\\
14,\\
\left( {x - 1} \right)\left( {x + 1} \right)\left( {x - 2} \right)\left( {x + 2} \right)\\
15,\\
\left( {{x^2} - 2x + 2} \right)\left( {{x^2} + 2x + 2} \right)
\end{array}\)
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
4,\\
2{x^2} - 5xy + 3{y^2}\\
= \left( {2{x^2} - 2xy} \right) + \left( { - 3xy + 3{y^2}} \right)\\
= 2x.\left( {x - y} \right) - 3y.\left( {x - y} \right)\\
= \left( {x - y} \right)\left( {2x - 3y} \right)\\
5,\\
xy + {y^2} - x - y\\
= \left( {xy + {y^2}} \right) + \left( { - x - y} \right)\\
= y\left( {x + y} \right) - \left( {x + y} \right)\\
= \left( {x + y} \right)\left( {y - 1} \right)\\
6,\\
25 - {x^2} + 4xy - 4{y^2}\\
= 25 - \left( {{x^2} - 4xy + 4{y^2}} \right)\\
= 25 - \left[ {{x^2} - 2.x.2y + {{\left( {2y} \right)}^2}} \right]\\
= {5^2} - {\left( {x - 2y} \right)^2}\\
= \left[ {5 - \left( {x - 2y} \right)} \right]\left[ {5 + \left( {x - 2y} \right)} \right]\\
= \left( {5 - x + 2y} \right)\left( {5 + x - 2y} \right)\\
7,\\
xy + xz - 2y - 2z\\
= \left( {xy + xz} \right) - \left( {2y + 2z} \right)\\
= x.\left( {y + z} \right) - 2.\left( {y + z} \right)\\
= \left( {y + z} \right).\left( {x - 2} \right)\\
8,\\
25{x^4} - 81{y^4}\\
= {\left( {5{x^2}} \right)^2} - {\left( {9{y^2}} \right)^2}\\
= \left( {5{x^2} - 9{y^2}} \right)\left( {5{x^2} + 9{y^2}} \right)\\
9,\\
{x^2} - 6xy + 9{y^2} - 25{z^2}\\
= \left( {{x^2} - 6xy + 9{y^2}} \right) - 25{z^2}\\
= \left[ {{x^2} - 2.x.3y + {{\left( {3y} \right)}^2}} \right] - {\left( {5z} \right)^2}\\
= {\left( {x - 3y} \right)^2} - {\left( {5z} \right)^2}\\
= \left[ {\left( {x - 3y} \right) - 5z} \right]\left[ {\left( {x - 3y} \right) + 5z} \right]\\
= \left( {x - 3y - 5z} \right)\left( {x - 3y + 5z} \right)\\
10,\\
3{x^2} - 3{y^2} - 12x + 12y\\
= 3.\left( {{x^2} - {y^2} - 4x + 4y} \right)\\
= 3.\left[ {\left( {{x^2} - {y^2}} \right) - \left( {4x - 4y} \right)} \right]\\
= 3.\left[ {\left( {x - y} \right)\left( {x + y} \right) - 4.\left( {x - y} \right)} \right]\\
= 3.\left( {x - y} \right)\left( {x + y - 4} \right)\\
11,\\
4{x^3} + 4x{y^2} + 8{x^2}y - 16x\\
= 4x.\left( {{x^2} + {y^2} + 2xy - 16} \right)\\
= 4x.\left[ {\left( {{x^2} + 2xy + {y^2}} \right) - 16} \right]\\
= 4x.\left[ {{{\left( {x + y} \right)}^2} - {4^2}} \right]\\
= 4x.\left[ {\left( {x + y} \right) - 4} \right].\left[ {\left( {x + y} \right) + 4} \right]\\
= 4x.\left( {x + y - 4} \right)\left( {x + y + 4} \right)\\
12,\\
{x^2} + 7 - 8x\\
= \left( {{x^2} - x} \right) + \left( { - 7x + 7} \right)\\
= x.\left( {x - 1} \right) - 7.\left( {x - 1} \right)\\
= \left( {x - 1} \right)\left( {x - 7} \right)\\
13,\\
{x^2} + 9x + 20\\
= \left( {{x^2} + 4x} \right) + \left( {5x + 20} \right)\\
= x\left( {x + 4} \right) + 5.\left( {x + 4} \right)\\
= \left( {x + 4} \right)\left( {x + 5} \right)\\
14,\\
{x^4} - 5{x^2} + 4\\
= \left( {{x^4} - {x^2}} \right) + \left( { - 4{x^2} + 4} \right)\\
= {x^2}\left( {{x^2} - 1} \right) - 4.\left( {{x^2} - 1} \right)\\
= \left( {{x^2} - 1} \right)\left( {{x^2} - 4} \right)\\
= \left( {{x^2} - {1^2}} \right)\left( {{x^2} - {2^2}} \right)\\
= \left( {x - 1} \right)\left( {x + 1} \right)\left( {x - 2} \right)\left( {x + 2} \right)\\
15,\\
{x^4} + 4\\
= \left( {{x^4} + 4{x^2} + 4} \right) - 4{x^2}\\
= \left[ {{{\left( {{x^2}} \right)}^2} + 2.{x^2}.2 + {2^2}} \right] - {\left( {2x} \right)^2}\\
= {\left( {{x^2} + 2} \right)^2} - {\left( {2x} \right)^2}\\
= \left[ {\left( {{x^2} + 2} \right) - 2x} \right].\left[ {\left( {{x^2} + 2} \right) + 2x} \right]\\
= \left( {{x^2} - 2x + 2} \right)\left( {{x^2} + 2x + 2} \right)
\end{array}\)
