simplify and express as a single fraction in its simplest form: (a) 4/a^2 -2a-3 + a/a-3 + a/a+1
I bet you mean:
4/(a^2 -2a-3) + a/(a-3) + a/(a+1)
= 4/( (a-3)(a+1) + a/(a-3) + a/(a+1)
= ( 4 + a(a+1) + a(a-3) )/( (a-3)(a+1) )
= (2a^2 - 2a + 4)/( (a-3)(a+1) )
or
= 2(a^2 - a + 2)/( a^2 - 2a - 3)
To simplify and express the given expression as a single fraction in its simplest form, we need to find a common denominator for all the fractions and then combine them.
Let's start by finding the common denominator. The denominators in the expression are a^2 - 2a - 3, a - 3, and a + 1.
To find the common denominator, we need to factorize each denominator:
a^2 - 2a - 3 = (a - 3)(a + 1)
a - 3 = (a - 3)
a + 1 = (a + 1)
Now, we can see that the common denominator is (a - 3)(a + 1).
Next, we can rewrite each fraction with the common denominator:
4/(a^2 - 2a - 3) + a/(a - 3) + a/(a + 1) = (4/(a - 3)(a + 1)) + (a/(a - 3)(a + 1)) + (a/(a - 3)(a + 1))
Now that all the fractions have the same denominator, we can combine them:
= (4 + a + a)/((a - 3)(a + 1))
Simplifying the numerator:
= (2a + 4)/((a - 3)(a + 1))
Finally, we have simplified and expressed the expression as a single fraction in its simplest form:
= (2(a + 2))/((a - 3)(a + 1))