1/4(y+b)+1/4(y-b)+y/2(y^2-b^2)-y^3/y^4-b^4
That is equal to
(-2b^4y - b^2y^2 + y^2 + y^4 - 2)/(2y)
I suspect you mean
1/(4(y+b))+1/(4(y-b))+y/(2(y^2-b^2))-y^3/(y^4-b^4) = (b^2y)/(y^4-b^4)
parentheses can make a huge difference. Use them.
b^2y/(y^4-b^4)
To simplify the given expression:
1/4(y+b) + 1/4(y-b) + y/2(y^2-b^2) - y^3/y^4-b^4
Step 1: Combine like terms in the numerator of each fraction.
1/4y + 1/4b + 1/4y - 1/4b + y/2(y^2-b^2) - y^3/y^4-b^4
Step 2: Simplify the fractions using a common denominator.
Common denominator for the first two fractions is 4.
(1/4y + 1/4b) + (1/4y - 1/4b) + y/2(y^2-b^2) - y^3/y^4-b^4
Step 3: Combine the like terms that have the same denominator.
(2/4y) + (2/4b) + y/2(y^2-b^2) - y^3/y^4-b^4
Step 4: Reduce the fractions, if possible.
(1/2y) + (1/2b) + y/2(y^2-b^2) - y^3/y^4-b^4
Step 5: Simplify the expression in the denominator of the third fraction.
The expression in the denominator of (y/2(y^2-b^2)) can be factored as y^2-b^2 = (y + b)(y - b).
So, the simplified expression is:
(1/2y) + (1/2b) + y/2(y + b)(y - b) - y^3/y^4-b^4
Remember, this is the simplified form of the expression you provided.