(1+3x)^6 Binomial Expansion

Jasonbradley

3 min read

·

Jun 16, 2024

27

Share

Understanding Binomial Expansion: (1 + 3x)^6

The binomial theorem provides a powerful tool for expanding expressions of the form (a + b)^n. In this article, we'll explore how to expand the binomial expression (1 + 3x)^6.

The Binomial Theorem

The binomial theorem states that:

(a + b)^n = ∑(n choose k) a^(n-k) b^k

where:

• n is a non-negative integer (the power)
• k is an integer ranging from 0 to n
• (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!), representing the number of ways to choose k items from a set of n items.

Expanding (1 + 3x)^6

Let's apply the binomial theorem to expand (1 + 3x)^6:

1. Identify a and b: In this case, a = 1 and b = 3x.

2. Set n: n = 6.

3. Calculate binomial coefficients: We'll need to calculate the binomial coefficients for k = 0, 1, 2, 3, 4, 5, and 6:

   • (6 choose 0) = 1
   • (6 choose 1) = 6
   • (6 choose 2) = 15
   • (6 choose 3) = 20
   • (6 choose 4) = 15
   • (6 choose 5) = 6
   • (6 choose 6) = 1

4. Apply the formula:

   (1 + 3x)^6 = (6 choose 0) 1^6 (3x)^0 + (6 choose 1) 1^5 (3x)^1 + (6 choose 2) 1^4 (3x)^2 + (6 choose 3) 1^3 (3x)^3 + (6 choose 4) 1^2 (3x)^4 + (6 choose 5) 1^1 (3x)^5 + (6 choose 6) 1^0 (3x)^6

5. Simplify:

   (1 + 3x)^6 = 1 + 18x + 135x^2 + 540x^3 + 1215x^4 + 1458x^5 + 729x^6

Therefore, the expansion of (1 + 3x)^6 is 1 + 18x + 135x^2 + 540x^3 + 1215x^4 + 1458x^5 + 729x^6.

Key Takeaways

• The binomial theorem provides a systematic way to expand expressions of the form (a + b)^n.
• Calculating binomial coefficients is crucial for the expansion.
• The expansion of (1 + 3x)^6 involves 7 terms, each with a specific coefficient and power of x.

By understanding and applying the binomial theorem, we can efficiently expand complex binomial expressions.