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 for problems 2, 3, and 4. Problem 2: Find the term independent of x in the expansion of (x - (3)/(x))^4. Step 1: Write the general term of the binomial expansion. The general term T_r+1 in the expansion of (a+b)^n is given by T_r+1 = nr a^n-r b^r. In this case, a=x, b=-(3)/(x) = -3x^-1, and n=4. T_r+1 = 4r (x)^4-r (-3x^-1)^r Step 2: Simplify the powers of x. T_r+1 = 4r x^4-r (-3)^r (x^-1)^r T_r+1 = 4r (-3)^r x^4-r x^-r T_r+1 = 4r (-3)^r x^4-2r Step 3: Find the value of r for the term independent of x. For the term to be independent of x, the exponent of x must be 0. 4 - 2r = 0 2r = 4 r = 2 Step 4: Substitute r=2 into the general term to find the term. T_2+1 = T_3 = 42 (-3)^2 x^4-2(2) T_3 = (4!)/(2!(4-2)!) (-3)^2 x^0 T_3 = (4 × 3 × 2 × 1)/((2 × 1)(2 × 1)) (9) × 1 T_3 = (24)/(4) (9) T_3 = 6 × 9 T_3 = 54 The term independent of x is 54. Problem 3: Find the first three terms in ascending powers of y in the expansion of (1-y)^(1)/(2). Step 1: Use the binomial expansion formula for fractional exponents. The formula for (1+x)^n is 1 + nx + (n(n-1))/(2!)x^2 + . Here, x = -y and n = (1)/(2). Step 2: Calculate the first term. The first term is 1. Step 3: Calculate the second term. nx = (1)/(2)(-y) = -(1)/(2)y Step 4: Calculate the third term. (n(n-1))/(2!)x^2 = (1)/(2)((1)/(2)-1)2!(-y)^2 = (1)/(2)(-(1)/(2))2 × 1 (y^2) = (-1)/(4)2 y^2 = -(1)/(8)y^2 Step 5: Combine the first three terms. The first three terms in ascending powers of y are 1 - (1)/(2)y - (1)/(8)y^2. The final answer is 1 - (1)/(2)y - (1)/(8)y^2. Problem 4: Find the coefficient of x^-1 in the expansion of ((x+1)(2x-(1)/(x))). Step 1: Expand the product of the two factors. (x+1)(2x-(1)/(x)) = x(2x) + x(-(1)/(x)) + 1(2x) + 1(-(1)/(x)) = 2x^2 - 1 + 2x - (1)/(x) Step 2: Rewrite the expression with negative exponents. = 2x^2 + 2x - 1 - x^-1 Step 3: Identify the term containing x^-1 and its coefficient. The term containing x^-1 is -x^-1. The coefficient of x^-1 is -1. The final answer is -1.