%5E3-expanded.jpeg "(3x-1)^3 Expanded")
Expanding (3x-1)^3
When working with algebraic expressions, we often come across situations where we need to expand expressions that involve parentheses raised to a power. One such example is (3x-1)^3. In this article, we will explore how to expand this expression using the binomial theorem.
The Binomial Theorem
The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (a+b)^n, where a and b are constants or variables, and n is a positive integer. The theorem states that:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + n(n-1)(n-2)...(2)(1)ab^(n-1) + b^n
Expanding (3x-1)^3
Now, let's apply the binomial theorem to expand (3x-1)^3. In this case, we have a = 3x and b = -1. Plugging these values into the formula, we get:
(3x-1)^3 = (3x)^3 - 3(3x)^2(1) + 3(3x)(1)^2 - (1)^3
Simplifying each term, we get:
(3x)^3 = 27x^3
- 3(3x)^2(1) = - 27x^2
3(3x)(1)^2 = 9x
- (1)^3 = -1
Now, we can combine these terms to get the final expanded form of (3x-1)^3:
(3x-1)^3 = 27x^3 - 27x^2 + 9x - 1
And that's it! We have successfully expanded (3x-1)^3 using the binomial theorem.
Conclusion
In this article, we have learned how to expand (3x-1)^3 using the binomial theorem. This technique is essential in algebra and is used to simplify complex expressions. By applying the binomial theorem, we can expand expressions of the form (a+b)^n and simplify them to their most basic form.
