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Use the Binomial Theorem to complete the expansion of (x+2)4(x + 2)^4.x4+8x3+24x2+32x+‾x^4 + 8x^3 + 24x^2 + 32x + \underline{\hspace{1cm}}

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Full solution

Q. Use the Binomial Theorem to complete the expansion of (x+2)4(x + 2)^4.x4+8x3+24x2+32x+‾x^4 + 8x^3 + 24x^2 + 32x + \underline{\hspace{1cm}}
  1. Identify general term: Identify the general term in the binomial expansion: T(k+1)=C(n,k)⋅a(n−k)⋅bkT(k+1) = C(n, k) \cdot a^{(n-k)} \cdot b^k where n=4n=4, a=xa=x, and b=2b=2.
  2. Calculate missing term: Calculate the missing term, which is the constant term when k=4k=4: T(5)=C(4,4)⋅x(4−4)⋅24T(5) = C(4, 4) \cdot x^{(4-4)} \cdot 2^4.
  3. Perform calculations: Perform the calculations: C(4,4)=1C(4, 4) = 1, x0=1x^0 = 1, and 24=162^4 = 16. So, T(5)=1×1×16=16T(5) = 1 \times 1 \times 16 = 16.

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