(2x-3)^4 Binomial Expansion

Jasonbradley

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Jun 16, 2024

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Binomial Expansion of (2x - 3)^4

The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a positive integer. Let's apply this to expand (2x - 3)^4.

Understanding the Binomial Theorem

The binomial theorem states:

(a + b)^n = ∑(n choose k) * a^(n-k) * b^k

where:

• n is the power to which the binomial is raised.
• k is an integer ranging from 0 to n.
• (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

Applying the Theorem to (2x - 3)^4

1. Identify a and b: In our case, a = 2x and b = -3.

2. Calculate the binomial coefficients: We need the binomial coefficients for n = 4. These are:

   • (4 choose 0) = 1
   • (4 choose 1) = 4
   • (4 choose 2) = 6
   • (4 choose 3) = 4
   • (4 choose 4) = 1

3. Substitute and expand: Now, we substitute the values into the binomial theorem formula:

   (2x - 3)^4 = (4 choose 0) * (2x)^4 * (-3)^0 + (4 choose 1) * (2x)^3 * (-3)^1 + (4 choose 2) * (2x)^2 * (-3)^2 + (4 choose 3) * (2x)^1 * (-3)^3 + (4 choose 4) * (2x)^0 * (-3)^4

4. Simplify:

   (2x - 3)^4 = 1 * 16x^4 * 1 + 4 * 8x^3 * (-3) + 6 * 4x^2 * 9 + 4 * 2x * (-27) + 1 * 1 * 81

   (2x - 3)^4 = 16x^4 - 96x^3 + 216x^2 - 216x + 81

Conclusion

Therefore, the binomial expansion of (2x - 3)^4 is 16x^4 - 96x^3 + 216x^2 - 216x + 81.

This method can be used to expand any binomial expression raised to a positive integer power. Remember to carefully calculate the binomial coefficients and substitute the values correctly.