Phân tích đa thức thành nhân tử
căn ab - căn a - căn b + 1
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\(\sqrt{x}+\sqrt{y}+\sqrt{xy}+1\)
\(=\sqrt{x}+\sqrt{y}+\sqrt{x}.\sqrt{y}+1\)
\(=\sqrt{x}\left(\sqrt{y}+1\right)+\left(\sqrt{y}+1\right)\)
\(=\left(\sqrt{x}+1\right)\left(\sqrt{y}+1\right)\)
\(x\sqrt{y}-y\sqrt{x}=\sqrt{x^2}.\sqrt{y}-\sqrt{y^2}.\sqrt{x}=\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\left(x\sqrt{y}-y\sqrt{x}\right)+\left(x-y\right)\)
\(=\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)+\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{xy}+\sqrt{x}+\sqrt{y}\right)\)
b, \(a+b+2\sqrt{a.b}=\sqrt{a^2}+\sqrt{b^2}+2\sqrt{ab}=\left(\sqrt{a}+\sqrt{b}\right)^2\) ( Vì a, b >= 0 )
c, \(a+b-2\sqrt{a.b}=\sqrt{a^2}+\sqrt{b^2}-2\sqrt{ab}=\left(\sqrt{a}-\sqrt{b}\right)^2\)( Vì a, b >= 0 )
\(12-\sqrt{x}-x=12-4\sqrt{x}+3\sqrt{x}-x=4\left(3-\sqrt{x}\right)+\sqrt{x}\left(3-\sqrt{x}\right)=\left(4+\sqrt{x}\right)\left(3-\sqrt{x}\right)\)
\(x-y-\sqrt{x}-\sqrt{y}\\ =x-y-\left(\sqrt{x}+\sqrt{y}\right)\\ =\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)-\left(\sqrt{x}+\sqrt{y}\right)\\ =\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}-1\right)\)
=(x-y)-(căn x+căn y)
=(căn x-căn y)(căn x+căn y)-(căn x+căn y)
=(căn x+căn y)(căn x-căn y-1)
\(=\sqrt{a}\left(\sqrt{b}-1\right)-\left(\sqrt{b}-1\right)\)
\(=\left(\sqrt{b}-1\right)\left(\sqrt{a}-1\right)\)
\(\sqrt{ab}-\sqrt{a}-\sqrt{b}+1\)
\(=\left(\sqrt{ab}-\sqrt{a}\right)-\left(\sqrt{b}-1\right)\)
\(=\sqrt{a}\left(\sqrt{b}-1\right)-\left(\sqrt{b}-1\right)\)
\(=\left(\sqrt{b}-1\right)\left(\sqrt{a}-1\right)\)