%5E4-expand.jpeg "(2x-3)^4 Expand")
Expanding (2x - 3)^4
The expression (2x - 3)^4 represents the product of (2x - 3) multiplied by itself four times. Expanding this expression can be done using the Binomial Theorem or by repeated multiplication.
Using the Binomial Theorem
The Binomial Theorem states that:
(a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)ab^(n-1) + b^n
where (n choose k) represents the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Applying this to our expression:
- a = 2x
- b = -3
- n = 4
We get:
(2x - 3)^4 = (2x)^4 + (4 choose 1)(2x)^3(-3) + (4 choose 2)(2x)^2(-3)^2 + (4 choose 3)(2x)(-3)^3 + (-3)^4
Calculating the binomial coefficients and simplifying:
(2x - 3)^4 = 16x^4 - 96x^3 + 216x^2 - 216x + 81
Using Repeated Multiplication
We can also expand the expression by repeatedly multiplying (2x - 3) by itself:
(2x - 3)^4 = (2x - 3)(2x - 3)(2x - 3)(2x - 3)
First, multiply the first two factors:
(2x - 3)(2x - 3) = 4x^2 - 12x + 9
Then, multiply the result by the third factor:
(4x^2 - 12x + 9)(2x - 3) = 8x^3 - 24x^2 + 18x - 12x^2 + 36x - 27 = 8x^3 - 36x^2 + 54x - 27
Finally, multiply the result by the last factor:
(8x^3 - 36x^2 + 54x - 27)(2x - 3) = 16x^4 - 72x^3 + 108x^2 - 54x - 24x^3 + 108x^2 - 162x + 81 = 16x^4 - 96x^3 + 216x^2 - 216x + 81
Conclusion
Both methods lead to the same result:
(2x - 3)^4 = 16x^4 - 96x^3 + 216x^2 - 216x + 81
Choosing the method depends on your preference and the specific problem you are solving. The Binomial Theorem provides a more concise and general approach, while repeated multiplication is more intuitive for smaller exponents.