Question

The factors of the expression $a+b+c+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca}$ are ____________.

Answer 1

$$\begin{gathered} a+b+c+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca} \\ ={\left(\sqrt{a}\right)}^2+{\left(\sqrt{b}\right)}^2+{\left(-\sqrt{c}\right)}^2+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca} \\ ={\left(\sqrt{a}\right)}^2+{\left(\sqrt{b}\right)}^2+{\left(-\sqrt{c}\right)}^2+2\sqrt{a}\sqrt{b}+2\sqrt{b}\left(-\sqrt{c}\right)+2\sqrt{a}\left(-\sqrt{c}\right) \\ ={\left(\sqrt{a}+\sqrt{b}-\sqrt{c}\right)}^2\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:a^2+b^2+c^2+2ab+2bc+2ac={\left(a+b+c\right)}^2\right) \\ =\left(\sqrt{a}+\sqrt{b}-\sqrt{c}\right)\left(\sqrt{a}+\sqrt{b}-\sqrt{c}\right) \end{gathered}$$

Hence, the factors of the expression $a+b+c+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca}$ are $\underline{\left(\sqrt{a}+\sqrt{b}-\sqrt{c}\right)\mathrm{and}\left(\sqrt{a}+\sqrt{b}-\sqrt{c}\right)}.$.