Find the lowest common denominator for each of the following expressions. Do not state
the restrictions or simplify. Show all your work.
2𝑥+1 / 𝑥^2−5𝑥−14 − 6𝑥 / 𝑥^2+3𝑥+2
To find the lowest common denominator for the two fractions, we need to find the least common multiple (LCM) of the denominators.
For the first fraction, the denominator is 𝑥^2−5𝑥−14. This quadratic expression can be factored as (𝑥−7)(𝑥+2).
For the second fraction, the denominator is 𝑥^2+3𝑥+2. This quadratic expression can be factored as (𝑥+1)(𝑥+2).
Now we have the factors of the denominators: (𝑥−7)(𝑥+2) and (𝑥+1)(𝑥+2).
To find the LCM, we need to take the highest power of each factor. So the LCM is (𝑥−7)(𝑥+1)(𝑥+2).
Therefore, the lowest common denominator for the two fractions is (𝑥−7)(𝑥+1)(𝑥+2).