Question

What is the least common denominator of the rational expressions below?

Answer 1

The first thing we need to do is factor the denominators in the rational expressions.
Notice that in the denominator of the first rational expression we have a common factor $x$, so we can factor x out:
$x^{2} +7x=x(x+7)$
Now, the denominator of the second rational expression is a quadratic polynomial, we can factor it by finding tow numbers whose product will 21 and its sum will be 10. Those numbers are 7 and 3, (7x3=21 and 7+3=10):
$x^{2} +10x+21=(x+7)(x+3)$

Lets rewrite our rational expression with our factored denominators:
$\frac{6}{ x(x+7)} - \frac{8}{(x+7)(x+3)}$

Now, to find the least common denominator we are going to take one of the common factors (x+7) and all the non-common factors: x and (x+3):
$x(x+7)(x+3)$

We can conclude that the correct answer is a. x(x+7)(x+3)