Final answer:
To express \(\frac{3}{{x^2 - 2x - 15}}\) with the lowest common denominator, factor the quadratic expression to get \(\frac{3}{{(x - 5)(x + 3)}}\). This is already the simplest form since the denominator is fully factored and no common factors exist with the numerator.
Step-by-step explanation:
To write the rational expression \(\frac{3}{{x^2 - 2x - 15}}\) with the lowest common denominator, we first need to factor the denominator. The expression x^2 - 2x - 15 can be factored into (x - 5)(x + 3). Thus, the factored form of the expression is \(\frac{3}{{(x - 5)(x + 3)}}\).
This is the simplest form of the expression because the denominator is already fully factored and there are no common factors between the numerator and denominator that can be cancelled out. Therefore, \(\frac{3}{{x^2 - 2x - 15}}\) is equivalent to \(\frac{3}{{(x - 5)(x + 3)}}\) and no further simplification or finding a lower common denominator is necessary, as it is already in its most simplified form.