%5E4-expanded.jpeg "(2m-1)^4 Expanded")
Expanding (2m-1)^4
Expanding the expression (2m-1)^4 involves applying the binomial theorem or using repeated multiplication. Let's explore both methods.
Using the Binomial Theorem
The binomial theorem states that for any real numbers a and b and any non-negative integer n:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
where k ranges from 0 to n, and (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!).
Applying this to our expression (2m-1)^4:
- a = 2m
- b = -1
- n = 4
We get:
(2m-1)^4 = Σ (4 choose k) * (2m)^(4-k) * (-1)^k
Expanding the sum for k from 0 to 4:
(4 choose 0) * (2m)^4 * (-1)^0 + (4 choose 1) * (2m)^3 * (-1)^1 + (4 choose 2) * (2m)^2 * (-1)^2 + (4 choose 3) * (2m)^1 * (-1)^3 + (4 choose 4) * (2m)^0 * (-1)^4
Calculating the binomial coefficients and simplifying:
1 * 16m^4 * 1 + 4 * 8m^3 * -1 + 6 * 4m^2 * 1 + 4 * 2m * -1 + 1 * 1 * 1
Finally, we obtain the expanded form:
(2m-1)^4 = 16m^4 - 32m^3 + 24m^2 - 8m + 1
Using Repeated Multiplication
We can also expand (2m-1)^4 by repeatedly multiplying the expression by itself:
(2m-1)^4 = (2m-1)(2m-1)(2m-1)*(2m-1)
First, multiply the first two factors:
(2m-1)*(2m-1) = 4m^2 - 4m + 1
Then, multiply the result by the third factor:
(4m^2 - 4m + 1)*(2m-1) = 8m^3 - 12m^2 + 6m - 1
Finally, multiply the last factor:
(8m^3 - 12m^2 + 6m - 1)*(2m-1) = 16m^4 - 32m^3 + 24m^2 - 8m + 1
Both methods lead to the same expanded form: 16m^4 - 32m^3 + 24m^2 - 8m + 1. You can choose the method that you find easiest to apply.