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Use Pascal's Triangle to complete the expansion of (u+v)3(u + v)^3. \newlineu3+3u2v+‾uv2+v3u^3 + 3u^2v + \underline{\hspace{1cm}}uv^2 + v^3

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Pascal's triangle and the Binomial Theoremright chevron icon

Full solution

Q. Use Pascal's Triangle to complete the expansion of (u+v)3(u + v)^3. \newlineu3+3u2v+‾uv2+v3u^3 + 3u^2v + \underline{\hspace{1cm}}uv^2 + v^3
  1. Pascal's Triangle Coefficients: Pascal's Triangle for the third row is 1,3,3,11, 3, 3, 1. These are the coefficients for the expansion.
  2. Calculate Next Term: The second term is 3u2v3u^2v, so the third term will be the next coefficient (3)(3) times uu to the power of (3−2)(3 - 2) times vv to the power of 22. So, 3×u3−2×v23 \times u^{3-2} \times v^2.
  3. Calculate Third Term: Calculate the third term: 3×u1×v2=3uv23 \times u^1 \times v^2 = 3uv^2.
  4. Complete Expansion: The complete expansion is u3+3u2v+3uv2+v3u^3 + 3u^2v + 3uv^2 + v^3.

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