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 expansions of the given expressions: 1) Expand (x+y)^4 Step 1: Use the binomial theorem, which states that (a+b)^n = _k=0^n nk a^n-k b^k. For (x+y)^4, we have a=x, b=y, and n=4. The expansion will have n+1=5 terms. Step 2: Calculate the binomial coefficients 4k for k=0, 1, 2, 3, 4. 40 = 1 41 = 4 42 = (4 × 3)/(2 × 1) = 6 43 = 4 44 = 1 Step 3: Substitute the coefficients and terms into the binomial expansion formula. (x+y)^4 = 40x^4y^0 + 41x^3y^1 + 42x^2y^2 + 43x^1y^3 + 44x^0y^4 (x+y)^4 = 1x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 1y^4 The expanded form is x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. 2) Expand (3+y)^3 Step 1: Use the binomial theorem. For (3+y)^3, we have a=3, b=y, and n=3. The expansion will have n+1=4 terms. Step 2: Calculate the binomial coefficients 3k for k=0, 1, 2, 3. 30 = 1 31 = 3 32 = 3 33 = 1 Step 3: Substitute the coefficients and terms into the binomial expansion formula. (3+y)^3 = 303^3y^0 + 313^2y^1 + 323^1y^2 + 333^0y^3 Step 4: Simplify each term. (3+y)^3 = 1 · (27) · (1) + 3 · (9) · y + 3 · (3) · y^2 + 1 · (1) · y^3 (3+y)^3 = 27 + 27y + 9y^2 + y^3 The expanded form is 27 + 27y + 9y^2 + y^3. That's 2 down. 3 left today — send the next one.