%5E6-binomial-expansion.jpeg "(2-3x)^6 Binomial Expansion")
Binomial Expansion of (2-3x)^6
In algebra, binomial expansion is a method of expanding powers of a binomial expression, which consists of two terms. The binomial theorem provides a formula for expanding such expressions. In this article, we will focus on the binomial expansion of (2-3x)^6.
Binomial Theorem
The binomial theorem states that for any positive integer n and real numbers a and b:
$(a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k$
where {n \choose k} = \frac{n!}{k!(n-k)!} is the binomial coefficient.
Expanding (2-3x)^6
To expand (2-3x)^6, we can use the binomial theorem with a = 2 and b = -3x. Then, we have:
$(2-3x)^6 = \sum_{k=0}^6 {6 \choose k} 2^{6-k} (-3x)^k$
Simplifying the expression, we get:
$(2-3x)^6 = \sum_{k=0}^6 {6 \choose k} 2^{6-k} (-1)^k 3^k x^k$
Now, let's calculate the binomial coefficients and simplify the terms:
$(2-3x)^6 = {6 \choose 0} 2^6 - {6 \choose 1} 2^5 (3x) + {6 \choose 2} 2^4 (3x)^2 - {6 \choose 3} 2^3 (3x)^3 + {6 \choose 4} 2^2 (3x)^4 - {6 \choose 5} 2 (3x)^5 + {6 \choose 6} (3x)^6$
$(2-3x)^6 = 64 - 6(32)(3x) + 15(16)(9x^2) - 20(8)(27x^3) + 15(4)(81x^4) - 6(2)(243x^5) + 729x^6$
$(2-3x)^6 = 64 - 576x + 2160x^2 - 4320x^3 + 3888x^4 - 1458x^5 + 729x^6$
And that's the expanded form of (2-3x)^6!
Conclusion
In this article, we have applied the binomial theorem to expand the expression (2-3x)^6. The resulting expression is a sixth-degree polynomial in x, with coefficients that can be calculated using the binomial theorem. This process can be applied to any binomial expression, making it a powerful tool in algebra and mathematics.
