Factors of a + b + c + 2√ab − 2√bc − 2√ca
The factors of the expression \[ a+b+c+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca} \] are ____________
Solution
\[ a+b+c+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca} \]
\[ =(\sqrt{a}+\sqrt{b})^2-2\sqrt{c}(\sqrt{a}+\sqrt{b})+c \]
\[ =(\sqrt{a}+\sqrt{b})^2 -2(\sqrt{a}+\sqrt{b})\sqrt{c} +(\sqrt{c})^2 \]
\[ =(\sqrt{a}+\sqrt{b}-\sqrt{c})^2 \]
\[ \boxed{(\sqrt{a}+\sqrt{b}-\sqrt{c})^2} \]