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.
ScanSolve AI Answer
Here are the solutions to problems 1, 2, 3, and 4. 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, and 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 expanded expression by x^2. x^2(x-2y)^3 = x^2(x^3 - 6x^2y + 12xy^2 - 8y^3) = x^2 · x^3 - x^2 · 6x^2y + x^2 · 12xy^2 - x^2 · 8y^3 = x^5 - 6x^4y + 12x^3y^2 - 8x^2y^3 The binomial expansion 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_r+1 for the binomial expansion (a+b)^n = _r=0^n nr a^n-r b^r. Here, a=x, b=-(3)/(x) = -3x^-1, and n=4. T_r+1 = 4r (x)^4-r (-3x^-1)^r T_r+1 = 4r x^4-r (-3)^r (x^-1)^r T_r+1 = 4r (-3)^r x^4-r-r T_r+1 = 4r (-3)^r x^4-2r Step 2: For the term independent of x, the power of x must be 0. 4-2r = 0 2r = 4 r = 2 Step 3: Substitute r=2 back 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)!) (9) x^0 T_3 = (4 × 3)/(2 × 1) × 9 × 1