%5E6-binomial-expansion.jpeg "(1+3x)^6 Binomial Expansion")
Binomial Expansion of (1+3x)^6
Introduction
In algebra, binomial expansion is a mathematical operation that expresses a power of a binomial (a polynomial with two terms) as a sum of terms involving various powers of the individual terms. In this article, we will explore the binomial expansion of (1+3x)^6.
Binomial Theorem
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial. It states that:
(a+b)^n = ∑(n choose k) * a^(n-k) * b^k
where n is a positive integer, a and b are the terms of the binomial, and k ranges from 0 to n.
Expansion of (1+3x)^6
Using the binomial theorem, we can expand (1+3x)^6 as follows:
(1+3x)^6 = ∑(6 choose k) * 1^(6-k) * (3x)^k
Calculating the Expansion
To calculate the expansion, we need to evaluate the binomial coefficients (6 choose k) and the powers of 1 and 3x.
(6 choose 0) = 1, so the first term is 1^6 = 1
(6 choose 1) = 6, so the second term is 6 \* 1^5 \* (3x)^1 = 18x
(6 choose 2) = 15, so the third term is 15 \* 1^4 \* (3x)^2 = 135x^2
(6 choose 3) = 20, so the fourth term is 20 \* 1^3 \* (3x)^3 = 540x^3
(6 choose 4) = 15, so the fifth term is 15 \* 1^2 \* (3x)^4 = 1215x^4
(6 choose 5) = 6, so the sixth term is 6 \* 1^1 \* (3x)^5 = 1458x^5
(6 choose 6) = 1, so the seventh term is 1 \* 1^0 \* (3x)^6 = 729x^6
Final Expansion
Combining the terms, we get:
(1+3x)^6 = 1 + 18x + 135x^2 + 540x^3 + 1215x^4 + 1458x^5 + 729x^6
This is the binomial expansion of (1+3x)^6.
Conclusion
In this article, we have explored the binomial expansion of (1+3x)^6 using the binomial theorem. The expansion involves calculating the binomial coefficients and the powers of the individual terms, resulting in a sum of seven terms involving various powers of x.
