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
Step 1: Expand the binomial term (2x - (1)/(x))^3. We use the binomial theorem (a+b)^n = n0a^n b^0 + n1a^n-1b^1 + n2a^n-2b^2 + + nna^0 b^n. Here, a=2x, b=-(1)/(x)=-x^-1, and n=3. The terms are: • Term 1: 30(2x)^3(-x^-1)^0 = 1 · (8x^3) · 1 = 8x^3 • Term 2: 31(2x)^2(-x^-1)^1 = 3 · (4x^2) · (-x^-1) = -12x^2-1 = -12x • Term 3: 32(2x)^1(-x^-1)^2 = 3 · (2x) · (x^-2) = 6x^1-2 = 6x^-1 • Term 4: 33(2x)^0(-x^-1)^3 = 1 · 1 · (-x^-3) = -x^-3 So, the expansion of (2x - (1)/(x))^3 is 8x^3 - 12x + 6x^-1 - x^-3. Step 2: Multiply the expanded form by (x+1) and identify terms with x^-3. We need to find the coefficient of x^-3 in the product (x+1)(8x^3 - 12x + 6x^-1 - x^-3). This product can be written as: x(8x^3 - 12x + 6x^-1 - x^-3) + 1(8x^3 - 12x + 6x^-1 - x^-3) Let's examine the terms that will result in x^-3: From the first part, x(8x^3 - 12x + 6x^-1 - x^-3): • x · (8x^3) = 8x^4 • x · (-12x) = -12x^2 • x · (6x^-1) = 6x^1-1 = 6x^0 = 6 • x · (-x^-3) = -x^1-3 = -x^-2 None of these terms result in x^-3. From the second part, 1(8x^3 - 12x + 6x^-1 - x^-3): • 1 · (8x^3) = 8x^3 • 1 · (-12x) = -12x • 1 · (6x^-1) = 6x^-1 • 1 · (-x^-3) = -x^-3 This term results in x^-3, and its coefficient is -1. Step 3: Sum the coefficients of x^-3. The only term contributing to x^-3 is -x^-3 from the second part of the multiplication. Therefore, the coefficient of x^-3 is -1. The final answer is -1.