Question

Factorize: $a^2+2+\frac{1}{a^2}$

• A. $(-a+\frac{1}{a}{)}^2$
• B. $(a-\frac{1}{a}{)}^2$
• C. $(a+\frac{1}{a}{)}^2$
• D. $(a+\frac{2}{a}{)}^2$

Answer 1

The correct option is C $(a+\frac{1}{a}{)}^2$

$\Rightarrow a^2+2+\frac{1}{a^2}$

Writing 2 in the form of $\frac{a}{a}+\frac{a}{a}$ where $\frac{a}{a}$ means 1,
$\Rightarrow a^2+\frac{a}{a}+\frac{a}{a}+\frac{1}{a^2}$

$\Rightarrow (a^2+\frac{a}{a})+(\frac{a}{a}+\frac{1}{a^2})$
$\Rightarrow a(a+\frac{1}{a})+\frac{1}{a}(a+\frac{1}{a})$
$\Rightarrow (a+\frac{1}{a})(a+\frac{1}{a})$
$\Rightarrow (a+\frac{1}{a}{)}^2$

Hence, $a^2+2+\frac{1}{a^2}$ =$(a+\frac{1}{a}{)}^2$.