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. 1. Find the binomial expansion of x^2(x-2y)^3. Step 1: Expand (x-2y)^3 using the binomial theorem (a+b)^n = _k=0^n nk a^n-k b^k. Here a=x, b=-2y, n=3. (x-2y)^3 = 30x^3(-2y)^0 + 31x^2(-2y)^1 + 32x^1(-2y)^2 + 33x^0(-2y)^3 = 1 · x^3 · 1 + 3 · x^2 · (-2y) + 3 · x · (4y^2) + 1 · 1 · (-8y^3) = x^3 - 6x^2y + 12xy^2 - 8y^3 Step 2: Multiply the result by x^2. x^2(x^3 - 6x^2y + 12xy^2 - 8y^3) = x^5 - 6x^4y + 12x^3y^2 - 8x^2y^3 The final answer is x^5 - 6x^4y + 12x^3y^2 - 8x^2y^3. 2. Find the term independent of x in the expansion of (x - (3)/(x))^4. Step 1: Write the general term T_k+1 for the expansion of (a+b)^n. Here a=x, b=-(3)/(x), n=4. T_k+1 = 4k (x)^4-k (-(3)/(x))^k = 4k x^4-k (-3)^k x^-k = 4k (-3)^k x^4-2k Step 2: For the term independent of x, set the power of x to 0. 4-2k = 0 2k = 4 k=2 Step 3: Substitute k=2 into the general term. T_2+1 = 42 (-3)^2 x^4-2(2) = 6 · 9 · x^0 = 54 The final answer is 54. 3. Find the first three terms in ascending powers of y in the expansion of (1-y)^(1)/(3). Use the binomial expansion for fractional powers: (1+x)^n = 1 + nx + (n(n-1))/(2!)x^2 + Here x=-y and n=(1)/(3). Step 1: First term. 1 Step 2: Second term. n(-y) = (1)/(3)(-y) = -(1)/(3)y Step 3: Third term. (n(n-1))/(2!)(-y)^2 = (1)/(3)((1)/(3)-1)2 y^2 = (1)/(3)(-(2)/(3))2 y^2 = (-2)/(9)2 y^2 = -(1)/(9)y^2 The first three terms are 1 - (1)/(3)y - (1)/(9)y^2. 4. Find the coefficient of x^-3 in the expansion of ((x+1)(2x - (1)/(x))^3). Step 1: Expand (2x - (1)/(x))^3. Let a=2x and b=-(1)/(x). (2x - (1)/(x))^3 = 30(2x)^3 + 31(2x)^2(-(1)/(x)) + 32(2x)(-(1)/(x))^2 + 33(-(1)/(x))^3 = 1 · 8x^3 + 3 · 4x^2 · (-(1)/(x)) + 3 · 2x · ((1)/(x^2)) + 1 · (-(1)/(x^3)) = 8x^3 - 12x + (6)/(x) - (1)/(x^3) Step 2: Multiply the expansion by (x+1). (x+1)(8x^3 - 12x + (6)/(x) - (1)/(x^3)) = x(8x^3 - 12x + (6)/(x) - (1)/(x^3)) + 1(8x^3 - 12x + (6)/(x) - (1)/(x^3)) = (8x^4 - 12x^2 + 6 - x^-2) + (8x^3 - 12x + 6x^-1 - x^-3) Step 3: Identify the term with x^-3. The term with x^-3 is -x^-3. The coefficient of x^-3 is -1. The final answer is -1.