Question

Phân tích các đa thức sau thành nhân tử:

a) \(3-\sqrt{3}+\sqrt{15}-3\sqrt{5}\) b) \(\sqrt{1-a}+\sqrt{1-a^2}\) với -1< a <1

c) \(\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}\) với a > 0, b > 0

d) \(x-y+\sqrt{xy^2}-\sqrt{y^3}\) với x > 0, y > 0

Answer 1

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)\(\sqrt{1-a}+\sqrt{1-a^2}=\sqrt{1-a}.1+\sqrt{1-a}.\sqrt{1+a}=\sqrt{1-a}\left(\sqrt{1+a}+1\right)\)

Answer 2

c)\(\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)+\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b+\sqrt{ab}\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(a+2\sqrt{ab}+b\right)=\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)^2\)

Answer 3

d) vì y>0 nên \(\sqrt{y^2}=y\)\(x-y+\sqrt{xy^2}-\sqrt{y^3}=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)+\sqrt{y^2}\left(\sqrt{x}-\sqrt{y}\right)=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}+y\right)\)