(2m-1)^4

3 min read Jun 16, 2024
(2m-1)^4

Expanding (2m-1)^4

The expression (2m-1)^4 represents the fourth power of the binomial (2m-1). To expand this, we can use the binomial theorem or simply multiply the expression by itself four times.

Using the Binomial Theorem

The binomial theorem states that:

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where:

  • n is the power
  • k ranges from 0 to n
  • (n choose k) is the binomial coefficient, calculated as n!/(k! * (n-k)!)

Applying this to our problem:

  • a = 2m
  • b = -1
  • n = 4

Therefore, we have:

(2m - 1)^4 = Σ (4 choose k) * (2m)^(4-k) * (-1)^k

This results in the following expansion:

(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

Simplifying the binomial coefficients and powers:

1 * 16m^4 * 1 + 4 * 8m^3 * -1 + 6 * 4m^2 * 1 + 4 * 2m * -1 + 1 * 1 * 1

Finally, we get the expanded form:

**(2m - 1)^4 = ** 16m^4 - 32m^3 + 24m^2 - 8m + 1

Expanding by Multiplication

We can also expand (2m-1)^4 by multiplying the expression by itself four times:

(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 this result by the third factor:

(4m^2 - 4m + 1) * (2m - 1) = 8m^3 - 12m^2 + 6m - 1

Finally, multiply this result by the last factor:

(8m^3 - 12m^2 + 6m - 1) * (2m - 1) = 16m^4 - 32m^3 + 24m^2 - 8m + 1

As we can see, both methods lead to the same expanded form: 16m^4 - 32m^3 + 24m^2 - 8m + 1.

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