Answer:
The least common denominator is (w - 1)×(w + 2)×(w + 5)
Step-by-step explanation:
The least common denominator is the lowest multiplication of the denominators which will allow the addition of the numerators of the fraction directly
The expression is given as follows;
(4w - 5)/(w² + w - 2) + w²/(w² + 7w + 10)
We factorize the denominator as follows;
For the denominator, (w² + w - 2), we have;
w² + w - 2 = 0
w² + w + 1/2² = 2 - 1/2²
(w + 1/2)² = 9/4
w + 1/2 = ±3/2
w = 1 or -2 Which gives;
w² + w - 2 = (w - 1)(w + 2)
For the denominator, (w² + 7w + 10), we have;
w² + 7w + 10 = 0
w² + 7w = - 10
w² + 7w + (7/2)² = - 10 + (7/2)² = 9/4
(w + 7/2)² = 9/4
w + 7/2 = ±3/2
w = -5 or -2
Which gives;
w² + 7w + 10 = (w + 5)(w + 2)
Therefore, we have;
(4w - 5)/(w² + w - 2) + w²/(w² + 7w + 10) = (4w - 5)/(w - 1)(w + 2) + w²/(w + 5)(w + 2)
Therefore, the factors are;
(w - 1),(w + 2) and (w + 5),(w + 2)
The least common denominator is found by multiplying each of the factors the the highest number of times they occur in each of the term which gives the least common denominator as (w - 1)×(w + 2)×(w + 5)
