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: We need to simplify the expression (1)/(x-y) - (1)/(x+y) + (2x)/(x^2-y^2). Step 1: Combine the first two fractions by finding a common denominator, which is (x-y)(x+y) = x^2-y^2. (1)/(x-y) - (1)/(x+y) = (1 · (x+y))/((x-y)(x+y)) - (1 · (x-y))/((x+y)(x-y)) = (x+y - (x-y))/(x^2-y^2) = (x+y-x+y)/(x^2-y^2) = (2y)/(x^2-y^2) Step 2: Add the result from Step 1 to the third fraction. (2y)/(x^2-y^2) + (2x)/(x^2-y^2) = (2y+2x)/(x^2-y^2) Step 3: Factor the numerator and the denominator. (2(y+x))/((x-y)(x+y)) = (2(x+y))/((x-y)(x+y)) Step 4: Cancel out the common term (x+y). = (2)/(x-y) The simplified expression is (2)/(x-y). Problem 2: We need to simplify the expression (x+2)/(1+x+x^2) - (x-2)/(1-x+x^2) - (2x^2)/(1+x^2+x^4). Step 1: Find a common denominator for the first two fractions. The product of their denominators is (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. ((x+2)(1-x+x^2) - (x-2)(1+x+x^2))/((1+x+x^2)(1-x+x^2)) = ((x-x^2+x^3+2-2x+2x^2) - (x+x^2+x^3-2-2x-2x^2))/(1+x^2+x^4) = ((x^3+x^2-x+2) - (x^3-x^2-x-2))/(1+x^2+x^4) Step 2: Simplify the numerator. (x^3+x^2-x+2 - x^3+x^2+x+2)/(1+x^2+x^4) = ((x^3-x^3) + (x^2+x^2) + (-x+x) + (2+2))/(1+x^2+x^4) = (2x^2+4)/(1+x^2+x^4) Step 3: Subtract the third fraction from the result of Step 2. (2x^2+4)/(1+x^2+x^4) - (2x^2)/(1+x^2+x^4) = (2x^2+4-2x^2)/(1+x^2+x^4) = (4)/(1+x^2+x^4) The simplified expression is (4)/(1+x^2+x^4). Problem 3: We need to simplify the expression (1)/(b+c) + (b+c)/(b^2-c^2). Step 1: Simplify the second term by factoring the denominator b^2-c^2 = (b-c)(b+c). (b+c)/(b^2-c^2) = (b+c)/((b-c)(b+c)) = (1)/(b-c) Step 2: Add the first term to the simplified second term. (1)/(b+c) + (1)/(b-c) Step 3: Find a common denominator, which is (b+c)(b-c) = b^2-c^2. = (1 · (b-c))/((b+c)(b-c)) + (1 · (b+c))/((b-c)(b+c)) = (b-c+b+c)/(b^2-c^2) Step 4: Simplify the numerator. = (2b)/(b^2-c^2) The simplified expression is (2b)/(b^2-c^2). Send me the next one 📸