Câu 26: Phân tích thành nhân tử:
a. \({x^2} – 9\)
b. \(4{x^2} – 25\)
c. \({x^6} – {y^6}\)
a. \({x^2} – 9)\\( = {x^2} – {3^2} = \left( {x + 3} \right)\left( {x – 3} \right)\)
b. \(4{x^2} – 25\) \( = {\left( {2x} \right)^2} – {5^2} = \left( {2x + 5} \right)\left( {2x – 5} \right)\)
c. \({x^6} – {y^6}\)
\(\eqalign{ & = {\left( {{x^3}} \right)^2} – {\left( {{y^3}} \right)^2} = \left( {{x^3} + {y^3}} \right)\left( {{x^3} – {y^3}} \right) \cr & = \left( {x + y} \right)\left( {{x^2} – xy + y} \right)\left( {x – y} \right)\left( {{x^2} + xy + {y^2}} \right) \cr} \)
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Câu 27: Phân tích thành nhân tử
a. \(9{x^2} + 6xy + {y^2}\)
b. \(6x – 9 – {x^2}\)
c. \({x^2} + 4{y^2} + 4xy\)
a. \(9{x^2} + 6xy + {y^2}\) \( = {\left( {3x} \right)^2} + 2.\left( {3x} \right)y + {y^2} = {\left( {3x + y} \right)^2}\)
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b. \(6x – 9 – {x^2}\) \( = – \left( {{x^2} – 2.x.3 + {3^2}} \right) = – {\left( {x – 3} \right)^2}\)
c. \({x^2} + 4{y^2} + 4xy\) \( = {x^2} + 2.x.\left( {2y} \right) + {\left( {2y} \right)^2} = {\left( {x + 2y} \right)^2}\)
Câu 28: Phân tích thành nhân tử
a. \({\left( {x + y} \right)^2} – {\left( {x – y} \right)^2}\)
b. \({\left( {3x + 1} \right)^2} – {\left( {x + 1} \right)^2}\)
c. \({x^3} + {y^3} + {z^3} – 3xyz\)
a. \({\left( {x + y} \right)^2} – {\left( {x – y} \right)^2}\) \( = \left[ {\left( {x + y} \right) + \left( {x – y} \right)} \right]\left[ {\left( {x + y} \right) – \left( {x – y} \right)} \right]\)
\( = \left( {x + y + x – y} \right)\left( {x + y – x + y} \right) = 2x.2y = 4xy\)
b. \({\left( {3x + 1} \right)^2} – {\left( {x + 1} \right)^2}\) \( = \left[ {\left( {3x + 1} \right) + \left( {x + 1} \right)} \right]\left[ {\left( {3x + 1} \right) – \left( {x + 1} \right)} \right]\)
\( = \left( {3x + 1 + x + 1} \right)\left( {3x + 1 – x – 1} \right) = \left( {4x + 2} \right).2x = 4x\left( {2x + 1} \right)\)
c. \({x^3} + {y^3} + {z^3} – 3xyz\) \( = {\left( {x + y} \right)^3} – 3xy\left( {x + y} \right) + {z^3} – 3xyz\)
\(\eqalign{ & = \left[ {{{\left( {x + y} \right)}^3} + {z^3}} \right] – 3xy\left( {x + y + z} \right) \cr & = \left( {x + y + z} \right)\left[ {{{\left( {x + y} \right)}^2} – \left( {x + y} \right)z + {z^2}} \right] – 3xy\left( {x + y + z} \right) \cr & = \left( {x + y + z} \right)\left( {{x^2} + 2xy + {y^2} – xz – yz + {z^2} – 3xy} \right) \cr & = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} – xy – xz – yz} \right) \cr} \)
