Verified Answer :

  • Identify the binomial components as \(a = 2m\) and \(b = n\).
  • Apply the identity \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).
  • Calculate each term: \(8m^3\), \(12m^2n\), \(6mn^2\), and \(n^3\).
  • The final expanded form is \(\boxed{8m^3 + 12m^2n + 6mn^2 + n^3}\).

Explanation

  1. Identifying the Components and Applying the Identity

To expand the expression \((2m + n)^3\), we use the algebraic identity for the cube of a binomial, which states that for any two terms \(a\) and \(b\):
\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]
In this specific problem, we identify the components as \(a = 2m\) and \(b = n\). We will substitute these values into the identity and simplify the resulting expression.

  1. Calculating and Expanding the Terms

Now, we substitute \(a = 2m\) and \(b = n\) into the identity:
\[(2m + n)^3 = (2m)^3 + 3(2m)^2(n) + 3(2m)(n)^2 + (n)^3\]
Next, we calculate each term individually:

  1. First term: \((2m)^3 = 2^3 \times m^3 = 8m^3\)
  2. Second term: \(3(2m)^2(n) = 3(4m^2)(n) = 12m^2n\)
  3. Third term: \(3(2m)(n^2) = 6mn^2\)
  4. Fourth term: \(n^3 = n^3\)

Summing these results gives the expanded polynomial.

  1. Final Result

Combining all the terms, we get:
\[(2m + n)^3 = 8m^3 + 12m^2n + 6mn^2 + n^3\]
This is the expanded form of the original binomial expression. The final result is \(\boxed{8m^3 + 12m^2n + 6mn^2 + n^3}\).

Examples

Expanding binomials is a foundational skill in algebra used for simplifying complex expressions and solving polynomial equations. For example, if you are calculating the volume of a cube where the side length is a binomial expression like \((2m + n)\), the volume is \((2m + n)^3\). Knowing how to expand this helps you understand how the total volume is distributed among the different components of the side length, which is crucial in engineering, physics, and computer graphics for modeling 3D objects.

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