6-binomial-theorem.jpeg "(2x-3)6 Binomial Theorem")
Binomial Theorem: Expanding (2x-3)^6
In algebra, the binomial theorem is a powerful tool for expanding expressions of the form (a+b)^n, where a and b are real numbers and n is a positive integer. In this article, we will explore the expansion of (2x-3)^6 using the binomial theorem.
What is the Binomial Theorem?
The binomial theorem is a mathematical formula for expanding powers of a binomial expression, such as (a+b)^n. It states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, a and b are real numbers, and \binom{n}{k} is the binomial coefficient, which can be calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (2x-3)^6 using the Binomial Theorem
Now, let's apply the binomial theorem to expand (2x-3)^6. We can write:
$(2x-3)^6 = \sum_{k=0}^6 \binom{6}{k} (2x)^{6-k} (-3)^k$
Using the formula for the binomial coefficient, we can calculate the values of \binom{6}{k} as:
k |
\binom{6}{k} |
|---|---|
| 0 | 1 |
| 1 | 6 |
| 2 | 15 |
| 3 | 20 |
| 4 | 15 |
| 5 | 6 |
| 6 | 1 |
Now, let's expand the expression:
$(2x-3)^6 = (2x)^6 - 6(2x)^5(3) + 15(2x)^4(3)^2 - 20(2x)^3(3)^3 + 15(2x)^2(3)^4 - 6(2x)(3)^5 + (3)^6$
Simplifying the expression, we get:
$(2x-3)^6 = 64x^6 - 576x^5 + 2160x^4 - 4320x^3 + 3888x^2 - 1728x + 729$
Conclusion
In this article, we have successfully expanded the expression (2x-3)^6 using the binomial theorem. The binomial theorem is a powerful tool for expanding expressions of the form (a+b)^n, and it has many applications in algebra, combinatorics, and other areas of mathematics.
