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Use the Binomial Theorem to complete the expansion of (y+z)2(y + z)^2. \newliney□+2yz+z2y^\square + 2yz + z^2

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Full solution

Q. Use the Binomial Theorem to complete the expansion of (y+z)2(y + z)^2. \newliney□+2yz+z2y^\square + 2yz + z^2
  1. Binomial Theorem Explanation: The Binomial Theorem states that (a+b)n=∑k=0n(nk)⋅an−k⋅bk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} \cdot a^{n-k} \cdot b^k. For (y+z)2(y + z)^2, n=2n=2.
  2. First Term Calculation: First term: ((20)⋅y(2−0)⋅z0=1⋅y2⋅1=y2)(2 \choose 0) \cdot y^{(2-0)} \cdot z^0 = 1 \cdot y^2 \cdot 1 = y^2.
  3. Second Term Calculation: Second term: (21)⋅y2−1⋅z1=2⋅y⋅z=2yz\binom{2}{1} \cdot y^{2-1} \cdot z^1 = 2 \cdot y \cdot z = 2yz.
  4. Third Term Calculation: Third term: ((22)⋅y(2−2)⋅z2=1⋅1⋅z2=z2)(2 \choose 2) \cdot y^{(2-2)} \cdot z^2 = 1 \cdot 1 \cdot z^2 = z^2.
  5. Combining All Terms: Combine all terms: y2+2yz+z2y^2 + 2yz + z^2.

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