(2m+n)^3

Jasonbradley

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Jun 16, 2024

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Expanding (2m + n)^3

In mathematics, expanding a binomial expression like (2m + n)^3 involves applying the distributive property and simplifying the resulting terms. Let's break down the process step by step:

Understanding the Binomial Theorem

The binomial theorem provides a general formula for expanding expressions of the form (x + y)^n:

(x + y)^n = ∑_(k=0)^n (n choose k) x^(n-k) y^k

where (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

Expanding (2m + n)^3

To expand (2m + n)^3, we can use the binomial theorem or apply the distributive property directly. Here's the breakdown:

1. Apply the distributive property:

   (2m + n)^3 = (2m + n)(2m + n)(2m + n)

   First, multiply the first two factors:

   (2m + n)(2m + n) = 4m^2 + 4mn + n^2

   Now, multiply this result by the third factor:

   (4m^2 + 4mn + n^2)(2m + n) = 8m^3 + 8m^2n + 2mn^2 + 4m^2n + 4mn^2 + n^3

2. Combine like terms:

   8m^3 + 12m^2n + 6mn^2 + n^3

Final Result

Therefore, the expansion of (2m + n)^3 is 8m^3 + 12m^2n + 6mn^2 + n^3.

Key Takeaways

• Understanding the binomial theorem provides a powerful tool for expanding binomial expressions.
• The distributive property is fundamental for multiplying expressions with multiple terms.
• Combining like terms simplifies the expanded expression.