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Use the Binomial Theorem to complete the expansion of (r+s)2(r + s)^2. \newliner□+2rs+s2r^\square + 2rs + s^2

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Full solution

Q. Use the Binomial Theorem to complete the expansion of (r+s)2(r + s)^2. \newliner□+2rs+s2r^\square + 2rs + s^2
  1. Binomial Theorem Explanation: The Binomial Theorem states that (a+b)n=Σk=0n(nk)⋅a(n−k)⋅bk(a + b)^n = \Sigma_{k=0}^{n} \binom{n}{k} \cdot a^{(n-k)} \cdot b^k, where Σ\Sigma denotes the sum over kk from 00 to nn. For (r+s)2(r + s)^2, n=2n = 2. We need to find the coefficient when k=1k = 1.
  2. Calculate Coefficient: Calculate the coefficient using ＂22 choose 11＂ which is rac{2!}{1! * (2-1)!} = 2.
  3. Multiply Coefficient: Multiply this coefficient by r2−1×s1r^{2-1} \times s^1 to get the term with rsrs.\newline2×r2−1×s1=2×r×s=2rs2 \times r^{2-1} \times s^1 = 2 \times r \times s = 2rs.
  4. Complete Expansion: Now we have the complete expansion: r2+2rs+s2r^2 + 2rs + s^2.

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