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a) \(\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}\)
\(=a\sqrt{a}-b\sqrt{b}+a\sqrt{b}-b\sqrt{a}\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)-\left(\sqrt{a}-\sqrt{b}\right)\sqrt{ab}\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b-\sqrt{ab}\right)\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(a+b\right)\)
b) \(x-y+\sqrt{xy^2}-\sqrt{y^3}\)
\(=\left(x-y\right)+\left(y\sqrt{x}-y\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)+y\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}+y\right)\)
\(a)\) \(xy-y\sqrt{x}+\sqrt{x}-1\)
= \(y\sqrt{x}.(\sqrt{x}-1)+\sqrt{x}-1\)
=\((\sqrt{x}-1).(y\sqrt{x}+1)\).
\(b)\)\(\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{ay}\)
=\(\sqrt{a}.\sqrt{x}-\sqrt{b}.\sqrt{y}+\sqrt{b}.\sqrt{x}-\sqrt{a}.\sqrt{y}\)
=\(\sqrt{a}.\sqrt{x}+\sqrt{b}.\sqrt{x}-\sqrt{a}.\sqrt{y}-\sqrt{b}.\sqrt{y}\)
=\(\sqrt{x}.(\sqrt{a}+\sqrt{b})-\sqrt{y}.(\sqrt{a}+\sqrt{b})\)
=\((\sqrt{x}-\sqrt{y}).(\sqrt{a}+\sqrt{b})\).
\(c)\)\(\sqrt{a+b}+\sqrt{a^2-b^2}\)
=\(\sqrt{a+b}+\sqrt{(a+b).(a-b)}\)
=\(\sqrt{a+b}+\sqrt{a+b}.\sqrt{a-b}\)
=\(\sqrt{a+b}.\left(1+\sqrt{a-b}\right)\).
\(d)\) \(12-\sqrt{x}-x\)
=\(12-4\sqrt{x}+3\sqrt{x}-x\)
=\(4.\left(3-\sqrt{x}\right)+\sqrt{x}\left(3-\sqrt{x}\right)\)
=\(\left(3-\sqrt{x}\right).\left(4+\sqrt{3}\right)\).
Phân tích thành nhân tử:
a) \(xy-y\sqrt{x}+\sqrt{x}-1\)
b) \(\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{ay}\)
a> = \(y\sqrt{x}\left(\sqrt{x}-1\right)+\left(\sqrt{x}-1\right)=\left(\sqrt{x}-1\right)\left(y\sqrt{x}-1\right)\)
a) \(xy-y\sqrt{x}+\sqrt{x}-1\)
\(=y\sqrt{x}\left(\sqrt{x}-1\right)+\left(\sqrt{x}-1\right)\)
\(=\left(\sqrt{x}-1\right)\left(y\sqrt{x}+1\right)\)
b) \(\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{ay}\)
\(=\left(\sqrt{ax}-\sqrt{ay}\right)+\left(-\sqrt{by}+\sqrt{bx}\right)\)
\(=\sqrt{a}.\left(\sqrt{x}-\sqrt{y}\right)+\sqrt{b}.\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{a}+\sqrt{b}\right)\)
1) \(x\sqrt{x}+y\sqrt{y}=\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)\)
2) \(x-3=\left(\sqrt{x}-\sqrt{3}\right)\left(\sqrt{x}+\sqrt{3}\right)\)
3) \(a+b=a-\left(-b\right)=\left(\sqrt{a}-\sqrt{-b}\right)\left(\sqrt{a}+\sqrt{-b}\right)\)
p/s: chúc bạn học tốt
a, \(5+\sqrt{5}=\sqrt{5}\left(\sqrt{5}+1\right)\)
b, \(a-2\sqrt{a}=\sqrt{a}\left(\sqrt{a}-2\right)\)
c, \(x-\sqrt{xy}=\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)\)
d, \(x-y-\sqrt{x}-\sqrt{y}\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)-\left(\sqrt{x}+\sqrt{y}\right)\)
\(=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}-1\right)\)
a/ \(=\sqrt{a}\left(\sqrt{x}-\sqrt{y}\right)-\sqrt{b}\left(\sqrt{x}-\sqrt{y}\right)=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{a}-\sqrt{b}\right)\)
b/ \(=\sqrt{\left(a-b\right)\left(a+b\right)}-\sqrt{\left(a+b\right)\left(a^2-ab+b^2\right)}\)
\(=\sqrt{a+b}\left(\sqrt{\left(a-b\right)\left(a^2-ab+b^2\right)}\right)\)
c/ \(=\left(\sqrt{x}-\frac{3}{2}\right)^2-\frac{81}{4}=\left(\sqrt{x}-\frac{3}{2}-\frac{9}{2}\right)\left(\sqrt{x}-\frac{3}{2}+\frac{9}{2}\right)=\left(\sqrt{x}-6\right)\left(\sqrt{x}+3\right)\)
\(a.\sqrt{ax}+\sqrt{by}-\sqrt{ay}-\sqrt{bx}\\ =\left(\sqrt{ax}-\sqrt{ay}\right)-\left(\sqrt{bx}-\sqrt{by}\right)\\ =\sqrt{a}\left(\sqrt{x}-\sqrt{y}\right)-\sqrt{b}\left(\sqrt{x}-\sqrt{y}\right)\\ =\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{x}-\sqrt{y}\right)\)
\(b.\sqrt{a^2-b^2}-\sqrt{a^3+b^3}\\ =\sqrt{\left(a+b\right)\left(a-b\right)}-\sqrt{\left(a+b\right)\left(a^2-ab+b^2\right)}\\ =\sqrt{a+b}\left(\sqrt{a-b}-\sqrt{a^2-ab+b^2}\right)\)
\(c.x-3\sqrt{x}-18=x-6\sqrt{x}+3\sqrt{x}-18\\ =\sqrt{x}\left(\sqrt{x}-6\right)+3\left(\sqrt{x}-6\right)\\ =\left(\sqrt{x}+3\right)\left(\sqrt{x}-6\right)\)
\(d.x\sqrt{x}+4x-12\sqrt{x}-27=\left(\sqrt{x^3}-27\right)+\left(4x-12\sqrt{x}\right)\\ =\left(\sqrt{x}-3\right)\left(x+3\sqrt{x}+9\right)+4\sqrt{x}\left(\sqrt{x}-3\right)\\ =\left(\sqrt{x}-3\right)\left(x+3\sqrt{x}+9+4\sqrt{x}\right)\\ =\left(\sqrt{x}-3\right)\left(x+7\sqrt{x}+9\right)\)
(có gì sai mong mọi người góp ý)
a)\(3-\sqrt{3}+\sqrt{15}-3\sqrt{5}=\sqrt{3}\left(\sqrt{3}-1\right)-\sqrt{15}\left(\sqrt{3}-1\right)=\left(\sqrt{3}-1\right)\left(\sqrt{3}-\sqrt{15}\right)=\sqrt{3}\left(\sqrt{3}-1\right)\left(1-\sqrt{5}\right)\)\(\)b)\(\sqrt{1-a}+\sqrt{1-a^2}=\sqrt{1-a}.1+\sqrt{1-a}.\sqrt{1+a}=\sqrt{1-a}\left(\sqrt{1+a}+1\right)\)
c)\(\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)+\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b+\sqrt{ab}\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(a+2\sqrt{ab}+b\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)^2\)