(x+2)^2 Binomial Expansion

Jasonbradley

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Jun 16, 2024

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Understanding the Binomial Expansion of (x+2)^2

The binomial expansion is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n. In this case, we'll focus on expanding (x + 2)^2.

Applying the Binomial Theorem

The binomial theorem states:

(a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)ab^(n-1) + b^n

where (n choose k) represents the binomial coefficient, calculated as:

(n choose k) = n! / (k! * (n-k)!)

Let's apply this to our expression, (x + 2)^2:

• a = x
• b = 2
• n = 2

Now, let's break down the expansion step-by-step:

1. (n choose 0)a^n b^0 = (2 choose 0)x^2 * 2^0 = 1 * x^2 * 1 = x^2
2. (n choose 1)a^(n-1)b^1 = (2 choose 1)x^1 * 2^1 = 2 * x * 2 = 4x
3. (n choose 2)a^(n-2)b^2 = (2 choose 2)x^0 * 2^2 = 1 * 1 * 4 = 4

Therefore, the complete binomial expansion of (x + 2)^2 is:

(x + 2)^2 = x^2 + 4x + 4

Alternative Approach

Instead of using the binomial theorem, you can also expand (x + 2)^2 by directly applying the distributive property:

(x + 2)^2 = (x + 2)(x + 2)

Expanding the product:

x(x + 2) + 2(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4

This method confirms the same result obtained using the binomial theorem.

Conclusion

Understanding the binomial expansion is crucial for simplifying complex algebraic expressions. By applying the binomial theorem or using the distributive property, we can efficiently expand expressions like (x + 2)^2 and obtain the correct result.