Question

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

6/x^2+7x - 8/x^2+10x+21
a. x(x+7)(x+3)
b. x+7
c. x(x+7)^2(x+3)
d. x(x+3)

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:

`(6)/( x(x+7)) - (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)