%5E4-binomial-expansion.jpeg "(1-3x)^4 Binomial Expansion")
Binomial Expansion of (1-3x)^4
The binomial theorem is a powerful tool that allows us to expand expressions of the form (a + b)^n. In this case, we want to expand (1 - 3x)^4.
Understanding the Binomial Theorem
The binomial theorem states that:
**(a + b)^n = a^n + n*a^(n-1)b + (n(n-1)/2!) * a^(n-2)b^2 + ... + b^n
where:
- n is a positive integer (the power)
- n! represents the factorial of n (n! = n * (n-1) * (n-2) * ... * 2 * 1)
- (n(n-1)/2!) represents the binomial coefficient, which is also denoted as n choose 2 or nCr
Expanding (1 - 3x)^4
Let's apply the binomial theorem to our expression:
- Identify a and b: In our case, a = 1 and b = -3x.
- Determine n: The power is n = 4.
Now, we can plug these values into the binomial theorem formula:
(1 - 3x)^4 = 1^4 + (4 * 1^3 * (-3x)) + (4 * 3 / 2! * 1^2 * (-3x)^2) + (4 * 1 / 3! * 1 * (-3x)^3) + (-3x)^4
Simplifying each term:
- 1^4 = 1
- 4 * 1^3 * (-3x) = -12x
- (4 * 3 / 2! * 1^2 * (-3x)^2) = 54x^2
- (4 * 1 / 3! * 1 * (-3x)^3) = -108x^3
- (-3x)^4 = 81x^4
Finally, combining all the terms:
(1 - 3x)^4 = 1 - 12x + 54x^2 - 108x^3 + 81x^4
Conclusion
By applying the binomial theorem, we successfully expanded (1 - 3x)^4 to get 1 - 12x + 54x^2 - 108x^3 + 81x^4. This method can be used to expand any binomial expression raised to a positive integer power.