Unveiling the Power of the Binomial Theorem for Expanding (x - 2y)^3

The Binomial Theorem is a fundamental concept in algebra that allows us to expand expressions of the form (a b)^n. When it comes to expanding expressions like (x - 2y)^3, understanding the Binomial Theorem offers a systematic approach to derive the expansion correctly.

Understanding the Binomial Theorem Formula

The generic formula for the Binomial Theorem for any positive integer n is given by:

Generic Formula: (a b)^n Σ (n k) a^(n-k) b^k
Where (n k) represents the binomial coefficient, and the summation is from k 0 to n.

Applying the Binomial Theorem to (x - 2y)^3

To expand the expression (x - 2y)^3, we first identify a, b, and n. In this case, a x, b -2y, and n 3.

Let's start by writing out the formula explicitly:

xy^3 Σ (3 k) x^(3-k) (-2y)^k
Where the summation runs from k 0 to 3.

Step-by-Step Expansion Using Pascal's Triangle

Let's use Pascal's Triangle to find the binomial coefficients (3 k).

Pascal's Triangle Row 3:

1 2 1

1 3 3 1

The coefficients for each term are thus 1, 3, 3, and 1.

Expanding the Expression

Now, let's substitute the values into the formula:

(x - 2y)^3 Σ (3 k) x^(3-k) (-2y)^k

This expands to:

(x - 2y)^3 (1)x^3 (3)x^2(-2y) (3)x(-2y)^2 (1)(-2y)^3

Simplifying each term:

(x - 2y)^3 1x^3 3x^2(-2y) 3x(-2y)^2 1(-2y)^3

(x - 2y)^3 x^3 - 6x^2y 12xy^2 - 8y^3

Conclusion

The Binomial Theorem is a powerful tool in algebra, allowing us to efficiently expand expressions like (x - 2y)^3. By utilizing the formula and understanding the coefficients from Pascal's Triangle, we can derive the correct expansion and simplify complex expressions.

Key Takeaways:

The Binomial Theorem simplifies the expansion of algebraic expressions. The coefficients in the expansion can be found using Pascal's Triangle. Understanding and applying the Binomial Theorem can save time and effort in solving algebraic problems.