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
To expand (2x-y)^6, we use the binomial theorem, which states that (a+b)^n = _k=0^n nk a^n-k b^k. In this case, a = 2x, b = -y, and n = 6. Step 1: Write out the general form of the expansion. (2x-y)^6 = 60(2x)^6(-y)^0 + 61(2x)^5(-y)^1 + 62(2x)^4(-y)^2 + 63(2x)^3(-y)^3 + 64(2x)^2(-y)^4 + 65(2x)^1(-y)^5 + 66(2x)^0(-y)^6 Step 2: Calculate the binomial coefficients nk. 60 = 1 61 = 6 62 = (6 × 5)/(2 × 1) = 15 63 = (6 × 5 × 4)/(3 × 2 × 1) = 20 64 = 62 = 15 65 = 61 = 6 66 = 1 Step 3: Substitute the coefficients and powers into the expansion and simplify each term. Term 1: 1 · (2x)^6 · (-y)^0 = 1 · 64x^6 · 1 = 64x^6 Term 2: 6 · (2x)^5 · (-y)^1 = 6 · 32x^5 · (-y) = -192x^5y Term 3: 15 · (2x)^4 · (-y)^2 = 15 · 16x^4 · y^2 = 240x^4y^2 Term 4: 20 · (2x)^3 · (-y)^3 = 20 · 8x^3 · (-y^3) = -160x^3y^3 Term 5: 15 · (2x)^2 · (-y)^4 = 15 · 4x^2 · y^4 = 60x^2y^4 Term 6: 6 · (2x)^1 · (-y)^5 = 6 · 2x · (-y^5) = -12xy^5 Term 7: 1 · (2x)^0 · (-y)^6 = 1 · 1 · y^6 = y^6 Step 4: Combine all the terms to get the final expansion. (2x-y)^6 = 64x^6 - 192x^5y + 240x^4y^2 - 160x^3y^3 + 60x^2y^4 - 12xy^5 + y^6 Drop the next question.