Question

Phân tích thành thừa số các biểu thức sau:

a) \(1+\sqrt{3}+\sqrt{5}+\sqrt{15}\) b) \(\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}\) c) \(\sqrt{35}-\sqrt{15}+\sqrt{14}-\sqrt{6}\)

d) \(3+\sqrt{18}+\sqrt{3+\sqrt{8}}\) e) \(xy+y\sqrt{x}+\sqrt{x}+1\) g) \(3+\sqrt{x}+9-x\)

Answer 1

a)  \(1+\sqrt{3}+\sqrt{5}+\sqrt{15}\)

\(=\left(1+\sqrt{3}\right)+\sqrt{5}\left(1+\sqrt{3}\right)\)

\(=\left(1+\sqrt{3}\right)\left(1+\sqrt{5}\right)\)

b)  \(\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}\)

\(=\sqrt{5}\left(\sqrt{2}+\sqrt{3}\right)+\sqrt{7}\left(\sqrt{2}+\sqrt{3}\right)\)

\(=\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{5}+\sqrt{7}\right)\)

c)  \(\sqrt{35}-\sqrt{15}+\sqrt{14}-\sqrt{6}\)

\(=\sqrt{5}\left(\sqrt{7}-\sqrt{3}\right)+\sqrt{2}\left(\sqrt{7}-\sqrt{3}\right)\)

\(=\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{2}\right)\)

Answer 2

e)  \(xy+y\sqrt{x}+\sqrt{x}+1\)

\(=y\sqrt{x}\left(\sqrt{x}+1\right)+\left(\sqrt{x}+1\right)\)

\(=\left(\sqrt{x}+1\right)\left(y\sqrt{x}+1\right)\)

g)  \(3+\sqrt{x}+9-x\)

\(=\left(3+\sqrt{x}\right)+\left(3+\sqrt{x}\right)\left(3-\sqrt{x}\right)\)

\(=\left(3+\sqrt{x}\right)\left(4-\sqrt{x}\right)\)