Đề bài
Phân tích đa thức sau thành nhân tử:
a) \({x^2} - 9{y^2}\) ;
b) \({4 \over {25}}{x^2} - 36{y^2}\) ;
c) \({(x + 5)^2} - 16\) ;
d) \(49{a^2} - {(3a - b)^2}\) ;
e) \({(2x + 3)^2} - {(x - 7)^2}\) ;
f) \(4{(x - y)^2} - {(3x + y)^2}\) .
Vận dụng hằng đẳng thức \({A^2} \pm 2AB + {B^2} = {(A \pm B)^2}\)
Lời giải chi tiết
\(\eqalign{ & a)\,\,{x^2} - 9{y^2} = {x^2} - {\left( {3y} \right)^2} \cr & \,\,\,\,\,\, = \left( {x + 3y} \right)\left( {x - 3y} \right) \cr & b)\,\,{4 \over {25}}{x^2} - 36{y^2} = 4\left( {{1 \over {25}}{x^2} - 9{y^2}} \right) \cr & \,\,\,\,\, = 4\left[ {{{\left( {{1 \over 5}x} \right)}^2} - {{\left( {3y} \right)}^2}} \right] \cr & \,\,\,\, = 4\left( {{1 \over 5}x - 3y} \right)\left( {{1 \over 5}x + 3y} \right) \cr & c)\,\,{\left( {x + 5} \right)^2} - 16 = {\left( {x + 5} \right)^2} - {4^2} \cr & \,\,\,\, = \left( {x + 5 - 4} \right)\left( {x + 5 + 4} \right) \cr & \,\,\,\, = \left( {x + 1} \right)\left( {x + 9} \right) \cr & d)\,\,49{a^2} - {\left( {3a - b} \right)^2} \cr & \,\,\,\, = {\left( {7a} \right)^2} - {\left( {3a - b} \right)^2} \cr & \,\,\,\, = \left[ {7a - \left( {3a - b} \right)} \right]\left[ {7a + \left( {3a - b} \right)} \right] \cr & \,\,\,\, = \left( {7a - 3a + b} \right)\left( {7a + 3a - b} \right) \cr & \,\,\,\, = \left( {4a + b} \right)\left( {10a - b} \right) \cr & e)\,\,{\left( {2x + 3} \right)^2} - {\left( {x - 7} \right)^2} \cr & \,\,\,\, = \left[ {\left( {2x + 3} \right) - \left( {x - 7} \right)} \right]\left[ {\left( {2x + 3} \right) + \left( {x - 7} \right)} \right] \cr & \,\,\,\, = \left( {2x + 3 - x + 7} \right)\left( {2x + 3 + x - 7} \right) \cr & \,\,\,\, = \left( {x + 10} \right)\left( {3x - 4} \right) \cr & f)\,\,4{\left( {x - y} \right)^2} - {\left( {3x + y} \right)^2} \cr & \,\,\,\, = {\left[ {2\left( {x - y} \right)} \right]^2} - {\left( {3x + y} \right)^2} \cr & \,\,\,\, = {\left( {2x - 2y} \right)^2} - {\left( {3x + y} \right)^2} \cr & \,\,\,\, = \left[ {\left( {2x - 2y} \right) - \left( {3x + y} \right)} \right]\left[ {\left( {2x - 2y} \right) + \left( {3x + y} \right)} \right] \cr & \,\,\,\, = \left( {2x - 2y - 3x - y} \right)\left( {2x - 2y + 3x + y} \right) \cr & \,\,\,\, = \left( { - x - 3y} \right)\left( {5x - y} \right) \cr} \)
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