%5E4-binomial-expansion.jpeg "(1-3x)^4 Binomial Expansion")
Binomial Expansion of (1-3x)^4
In algebra, the binomial expansion is a mathematical operation that involves expanding an expression of the form (a+b)^n, where n is a positive integer. In this article, we will explore the binomial expansion of (1-3x)^4.
What is Binomial Expansion?
Binomial expansion is a method of expanding an expression of the form (a+b)^n, where n is a positive integer. It is based on the formula:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + … + nb^(n-1)a + b^n
This formula is known as the binomial theorem.
Binomial Expansion of (1-3x)^4
To expand (1-3x)^4, we can use the binomial theorem with a = 1 and b = -3x. The formula becomes:
(1-3x)^4 = 1^4 + 4(1)^3(-3x) + 6(1)^2(-3x)^2 + 4(1)(-3x)^3 + (-3x)^4
Simplifying the expression, we get:
(1-3x)^4 = 1 - 12x + 54x^2 - 108x^3 + 81x^4
Breaking Down the Expansion
Let's break down the expansion to understand how we got each term:
- 1: This is the first term of the expansion, which is simply 1^4.
- -12x: This is the second term, which is 4(1)^3(-3x) = 4(-3x) = -12x.
- 54x^2: This is the third term, which is 6(1)^2(-3x)^2 = 6(-9x^2) = 54x^2.
- -108x^3: This is the fourth term, which is 4(1)(-3x)^3 = 4(-27x^3) = -108x^3.
- 81x^4: This is the fifth and final term, which is (-3x)^4 = 81x^4.
Conclusion
In this article, we have explored the binomial expansion of (1-3x)^4. We have seen how to apply the binomial theorem to expand the expression and break down the expansion to understand how each term is derived. The final expansion is:
(1-3x)^4 = 1 - 12x + 54x^2 - 108x^3 + 81x^4
This expansion can be used in various mathematical and real-world applications, such as algebra, calculus, and statistics.
