Answer
(1) Factorize each expression separately.
(2) Multiply the factors they have in common by the factors they do not share.
Work Step by Step
The exercise asks us to use $x^2 - 100$ and $x^2 - 20x + 100$ as an example. The first expression is a difference of squares: $$x^2 - 100 = x^2 - 10^2$$ which is solved by the formula $(a^2 - b^2) = (a+b)(a-b)$: $$x^2 - 100 = (x+10)(x-10)$$
The second expression is a quadratic expression that can be factorized by finding two factors of $100$ that, when added, give $-20$. The factors that satisfy this condition are $-10\times-10$: $$x^2 - 20x + 100 = (x-10)(x-10)$$.
To find the common denominator, we identify the factors they have in common: one factor of $(x-10)$. Then, we identify the factors they don't have in common: the other factor of $(x-10)$ and the remaining $(x+10)$. We now multiply the factors in common by the factors they do not share:$$CommonDenominator = (x-10)(x-10)(x+10)$$
