This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — here are the solutions to the problems: 1) (1)/(x-y) - (1)/(x+y) + (2x)/(x^2-y^2) Step 1: Find a common denominator for the first two terms, which is (x-y)(x+y) = x^2-y^2. (1(x+y))/((x-y)(x+y)) - (1(x-y))/((x+y)(x-y)) + (2x)/(x^2-y^2) Step 2: Combine the first two fractions. ((x+y) - (x-y))/(x^2-y^2) + (2x)/(x^2-y^2) Step 3: Simplify the numerator of the first combined fraction. (x+y-x+y)/(x^2-y^2) + (2x)/(x^2-y^2) = (2y)/(x^2-y^2) + (2x)/(x^2-y^2) Step 4: Combine all fractions since they now have a common denominator. (2y+2x)/(x^2-y^2) Step 5: Factor out 2 from the numerator and factor the denominator using the difference of squares formula (x^2-y^2 = (x-y)(x+y)). (2(x+y))/((x-y)(x+y)) Step 6: Cancel out the common factor (x+y). (2)/(x-y) The simplified expression is (2)/(x-y). 2) (x+2)/(1+x+x^2) - (x-2)/(1+x+x^2) - (2x^2)/(1+x^2+x^4) Step 1: Combine the first two terms as they share a common denominator. ((x+2) - (x-2))/(1+x+x^2) - (2x^2)/(1+x^2+x^4) Step 2: Simplify the numerator of the first term. (x+2-x+2)/(1+x+x^2) - (2x^2)/(1+x^2+x^4) = (4)/(1+x+x^2) - (2x^2)/(1+x^2+x^4) Step 3: Factor the denominator of the second term using the identity a^4+a^2b^2+b^4 = (a^2+ab+b^2)(a^2-ab+b^2). Here, a=x and b=1. 1+x^2+x^4 = (x^2+x+1)(x^2-x+1) Substitute this into the expression: (4)/(1+x+x^2) - (2x^2)/((1+x+x^2)(1-x+x^2)) Step 4: Find a common denominator, which is (1+x+x^2)(1-x+x^2). (4(1-x+x^2))/((1+x+x^2)(1-x+x^2)) - (2x^2)/((1+x+x^2)(1-x+x^2)) Step 5: Combine the fractions. (4(1-x+x^2) - 2x^2)/((1+x+x^2)(1-x+x^2)) Step 6: Expand and simplify the numerator. (4-4x+4x^2 - 2x^2)/((1+x+x^2)(1-x+x^2)) = (4-4x+2x^2)/(1+x^2+x^4) Step 7: Factor out 2 from the numerator. (2(x^2-2x+2))/(x^4+x^2+1) The simplified expression is (2(x^2-2x+2))/(x^4+x^2+1). 3) (1)/(b+c) + (b-c)/(b^2-c^2) Step 1: Factor the denominator of the second term using the difference of squares formula (b^2-c^2 = (b-c)(b+c)). (1)/(b+c) + (b-c)/((b-c)(b+c)) Step 2: Cancel out the common factor (b-c) in the second term. (1)/(b+c) + (1)/(b+c) Step 3: Combine the fractions as they have a common denominator. (1+1)/(b+c) = (2)/(b+c) The simplified expression is (2)/(b+c). Drop the next question.