Phân tích các đa thức sau thành nhân tử:
\( \eqalign{& a)\,\,3x - 7\sqrt x + 4 \cr & b)\,x\sqrt x - x - 4 \cr & c)\,3x\sqrt x - 7x + 17\sqrt x - 5 \cr} \)
\( \eqalign{& d)\,\,x\sqrt x + 5x + 8\sqrt x + 4 \cr & e)\,\,x - \sqrt x - 2001.2002 \cr & f)\,2x\sqrt x + x - 5\sqrt x - 4 \cr} \)
Câu 204806: Phân tích các đa thức sau thành nhân tử:
\( \eqalign{& a)\,\,3x - 7\sqrt x + 4 \cr & b)\,x\sqrt x - x - 4 \cr & c)\,3x\sqrt x - 7x + 17\sqrt x - 5 \cr} \)
\( \eqalign{& d)\,\,x\sqrt x + 5x + 8\sqrt x + 4 \cr & e)\,\,x - \sqrt x - 2001.2002 \cr & f)\,2x\sqrt x + x - 5\sqrt x - 4 \cr} \)
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Giải chi tiết:
\( \eqalign{& a)\,3x - 7\sqrt x + 4 = 3x - 3\sqrt x - 4\sqrt x + 4 \cr & = 3\sqrt x \left( {\sqrt x - 1} \right) - 4\left( {\sqrt x - 1} \right) \cr & = \left( {\sqrt x - 1} \right)\left( {3\sqrt x - 4} \right). \cr & b)\,x\sqrt x - x - 4 = {\left( {\sqrt x } \right)^3} - x - 8 + 4 \cr & = \left[ {{{\left( {\sqrt x } \right)}^3} - 8} \right] - \left( {x - 4} \right) \cr & = \left( {\sqrt x - 2} \right)\left( {x + 2\sqrt x + 4} \right) - \left( {\sqrt x - 2} \right)\left( {\sqrt x + 2} \right) \cr & = \left( {\sqrt x - 2} \right)\left( {x + \sqrt x + 2} \right). \cr & c)\,3x\sqrt x - 7x + 17\sqrt x - 5 \cr & = 3x\sqrt x - x - 6x + 2\sqrt x + 15\sqrt x - 5 \cr & = x\left( {3\sqrt x - 1} \right) - 2\sqrt x \left( {3\sqrt x - 1} \right) + 5\left( {3\sqrt x - 1} \right) \cr & = \left( {3\sqrt x - 1} \right)\left( {x - 2\sqrt x + 5} \right). \cr} \)
\( \eqalign{& d)\,x\sqrt x + 5x + 8\sqrt x + 4 \cr & = x\sqrt x + x + 4x + 4\sqrt x + 4\sqrt x + 4 \cr & = x\left( {\sqrt x + 1} \right) + 4\sqrt x \left( {\sqrt x + 1} \right) + 4\left( {\sqrt x + 1} \right) \cr & = \left( {\sqrt x + 1} \right)\left( {x + 4\sqrt x + 4} \right) \cr & = \left( {\sqrt x + 1} \right){\left( {\sqrt x + 2} \right)^2}. \cr & e)\,x - \sqrt x - 2001.2002 \cr & = x - \sqrt x - 2001\left( {2001 + 1} \right) \cr & = x - \sqrt x - {2001^2} - 2001 \cr & = \left( {\sqrt x - 2001} \right)\left( {\sqrt x + 2001} \right) - \left( {\sqrt x + 2001} \right) \cr & = \left( {\sqrt x + 2001} \right)\left( {\sqrt x - 2002} \right). \cr & f)\,2x\sqrt x + x - 5\sqrt x - 4 \cr & = 2x\sqrt x + 2x - x - \sqrt x - 4\sqrt x - 4 \cr & = 2x\left( {\sqrt x + 1} \right) - \sqrt x \left( {\sqrt x + 1} \right) - 4\left( {\sqrt x + 1} \right) \cr & = \left( {\sqrt x + 1} \right)\left( {2x - \sqrt x - 4} \right). \cr} \)
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