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Find the product. Simplify your answer.\newline(2k+1)(2k−1)(2k + 1)(2k - 1)

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Full solution

Q. Find the product. Simplify your answer.\newline(2k+1)(2k−1)(2k + 1)(2k - 1)
  1. Identify aa and bb: We are given the expression (2k+1)(2k−1)(2k + 1)(2k - 1) and need to find its product.\newlineThis expression is in the form of (a+b)(a−b)(a + b)(a - b), which is a difference of squares.\newlineThe special case for the difference of squares is: (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.
  2. Apply difference of squares: Identify the values of aa and bb.\newlineCompare (2k+1)(2k−1)(2k + 1)(2k - 1) with (a+b)(a−b)(a + b)(a - b).\newlinea = 2k2k\newlineb = 11
  3. Expand and simplify: Apply the difference of squares formula to expand (2k+1)(2k−1)(2k + 1)(2k - 1).\newline(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2\newline(2k+1)(2k−1)=(2k)2−(1)2(2k + 1)(2k - 1) = (2k)^2 - (1)^2
  4. Expand and simplify: Apply the difference of squares formula to expand (2k+1)(2k−1)(2k + 1)(2k - 1).\newline(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2\newline(2k+1)(2k−1)=(2k)2−(1)2(2k + 1)(2k - 1) = (2k)^2 - (1)^2Simplify (2k)2−(1)2(2k)^2 - (1)^2.\newline(2k)2−(1)2=(2k⋅2k)−(1⋅1)(2k)^2 - (1)^2 = (2k \cdot 2k) - (1 \cdot 1)\newline=4k2−1= 4k^2 - 1

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