(2x-y)^5 Binomial Theorem

Jasonbradley

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Jun 16, 2024

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Expanding (2x - y)^5 using the Binomial Theorem

The binomial theorem provides a powerful way to expand expressions of the form (a + b)^n. Let's see how to apply it to expand (2x - y)^5.

The Binomial Theorem

The binomial theorem states:

(a + b)^n = ∑_(k=0)^n (n choose k) a^(n-k) b^k

where (n choose k) is the binomial coefficient, calculated as:

(n choose k) = n! / (k! * (n-k)!)

Applying the Theorem to (2x - y)^5

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

2. Determine n: n = 5.

3. Calculate the binomial coefficients:

   • (5 choose 0) = 1
   • (5 choose 1) = 5
   • (5 choose 2) = 10
   • (5 choose 3) = 10
   • (5 choose 4) = 5
   • (5 choose 5) = 1

4. Apply the formula:

(2x - y)^5 = (5 choose 0) (2x)^5 (-y)^0 + (5 choose 1) (2x)^4 (-y)^1 + (5 choose 2) (2x)^3 (-y)^2 + (5 choose 3) (2x)^2 (-y)^3 + (5 choose 4) (2x)^1 (-y)^4 + (5 choose 5) (2x)^0 (-y)^5

5. Simplify:

**(2x - y)^5 = ** 1 * 32x^5 * 1 + 5 * 16x^4 * (-y) + 10 * 8x^3 * y^2 + 10 * 4x^2 * (-y)^3 + 5 * 2x * y^4 + 1 * 1 * (-y)^5

(2x - y)^5 = 32x^5 - 80x^4y + 80x^3y^2 - 40x^2y^3 + 10xy^4 - y^5

Conclusion

Therefore, the expanded form of (2x - y)^5 using the binomial theorem is 32x^5 - 80x^4y + 80x^3y^2 - 40x^2y^3 + 10xy^4 - y^5. This method efficiently expands binomials raised to any power, providing a powerful tool in algebra and calculus.