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
Okay, Princess, let's expand (1-2x)^6. The binomial theorem states that for any positive integer n, the expansion of (a+b)^n is given by: (a+b)^n = _r=0^n C(n, r) a^n-r b^r In this case, a=1, b=-2x, and n=6. Step 1: Identify the values for a, b, and n. a = 1 b = -2x n = 6 Step 2: Calculate the binomial coefficients C(n, r) for r=0 to r=6. C(6, 0) = 1 C(6, 1) = 6 C(6, 2) = (6 × 5)/(2 × 1) = 15 C(6, 3) = (6 × 5 × 4)/(3 × 2 × 1) = 20 C(6, 4) = C(6, 2) = 15 C(6, 5) = C(6, 1) = 6 C(6, 6) = 1 Step 3: Calculate each term of the expansion. Term 1 (r=0): C(6, 0) (1)^6-0 (-2x)^0 = 1 · 1^6 · 1 = 1 Term 2 (r=1): C(6, 1) (1)^6-1 (-2x)^1 = 6 · 1^5 · (-2x) = -12x Term 3 (r=2): C(6, 2) (1)^6-2 (-2x)^2 = 15 · 1^4 · (4x^2) = 60x^2 Term 4 (r=3): C(6, 3) (1)^6-3 (-2x)^3 = 20 · 1^3 · (-8x^3) = -160x^3 Term 5 (r=4): C(6, 4) (1)^6-4 (-2x)^4 = 15 · 1^2 · (16x^4) = 240x^4 Term 6 (r=5): C(6, 5) (1)^6-5 (-2x)^5 = 6 · 1^1 · (-32x^5) = -192x^5 Term 7 (r=6): C(6, 6) (1)^6-6 (-2x)^6 = 1 · 1^0 · (64x^6) = 64x^6 Step 4: Sum all the terms to get the full expansion. (1-2x)^6 = 1 - 12x + 60x^2 - 160x^3 + 240x^4 - 192x^5 + 64x^6 The expansion of (1-2x)^6 is 1 - 12x + 60x^2 - 160x^3 + 240x^4 - 192x^5 + 64x^6. That's 2 down. 3 left today — send the next one.