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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Here are the solutions to the problems. Problem 1: Simplify the expression: (1)/(x-y) - (1)/(x+y) + (2x)/(x^2-y^2) Step 1: Find a common denominator. We know that x^2-y^2 = (x-y)(x+y). So, the common denominator for all terms is x^2-y^2. (1)/(x-y) = (1 · (x+y))/((x-y)(x+y)) = (x+y)/(x^2-y^2) (1)/(x+y) = (1 · (x-y))/((x+y)(x-y)) = (x-y)/(x^2-y^2) Substitute these into the original expression: (x+y)/(x^2-y^2) - (x-y)/(x^2-y^2) + (2x)/(x^2-y^2) Step 2: Combine the numerators over the common denominator. ((x+y) - (x-y) + 2x)/(x^2-y^2) Step 3: Simplify the numerator. (x+y-x+y+2x)/(x^2-y^2) (2y+2x)/(x^2-y^2) Step 4: Factor the numerator and simplify further. (2(y+x))/(x^2-y^2) Since x^2-y^2 = (x-y)(x+y), we can write: (2(x+y))/((x-y)(x+y)) Assuming x+y ≠ 0, we can cancel the (x+y) term. (2)/(x-y) Problem 2: Simplify the expression: (x+2)/(1+x+x^2) - (x-2)/(1-x+x^2) - (2x)/(1+x^2+x^4) Step 1: Identify the relationship between the denominators. Notice that (1+x+x^2)(1-x+x^2) = ( (1+x^2)+x ) ( (1+x^2)-x ) = (1+x^2)^2 - x^2 = 1+2x^2+x^4-x^2 = 1+x^2+x^4. This means the common denominator for the first two terms is the denominator of the third term. Step 2: Combine the first two fractions using the common denominator 1+x^2+x^4. ((x+2)(1-x+x^2))/((1+x+x^2)(1-x+x^2)) - ((x-2)(1+x+x^2))/((1-x+x^2)(1+x+x^2)) ((x+2)(1-x+x^2) - (x-2)(1+x+x^2))/(1+x^2+x^4) Step 3: Expand the numerators. First part: (x+2)(1-x+x^2) = x(1-x+x^2) + 2(1-x+x^2) = x-x^2+x^3+2-2x+2x^2 = x^3+x^2-x+2. Second part: (x-2)(1+x+x^2) = x(1+x+x^2) - 2(1+x+x^2) = x+x^2+x^3-2-2x-2x^2 = x^3-x^2-x-2. Step 4: Substitute the expanded numerators back and simplify. ((x^3+x^2-x+2) - (x^3-x^2-x-2))/(1+x^2+x^4) (x^3+x^2-x+2-x^3+x^2+x+2)/(1+x^2+x^4) (2x^2+4)/(1+x^2+x^4) Step 5: Combine this result with the third term of the original expression. (2x^2+4)/(1+x^2+x^4) - (2x)/(1+x^2+x^4) (2x^2+4-2x)/(1+x^2+x^4) Step 6: Rearrange the numerator and factor out 2. (2(x^2-x+2))/(1+x^2+x^4) Problem 3: Simplify the expression: (1)/(b+c) + (b+c)/(b^2-c^2) Step 1: Find a common denominator. We know that b^2-c^2 = (b-c)(b+c). So, the common denominator is b^2-c^2. (1)/(b+c) = (1 · (b-c))/((b+c)(b-c)) = (b-c)/(b^2-c^2) Substitute this into the original expression: (b-c)/(b^2-c^2) + (b+c)/(b^2-c^2) Step 2: Combine the numerators over the common denominator. ((b-c) + (b+c))/(b^2-c^2) Step 3: Simplify the numerator. (b-c+b+c)/(b^2-c^2) (2b)/(b^2-c^2) (2b)/(b^2-c^2) Send me the next one 📸