How to Expand 12y^3 Using Pascal’s Triangle

Pascal’s Triangle is an ancient mathematical tool used for various purposes, one of which is binomial expansion. This article will guide you through expanding the binomial 12y^3 using Pascal’s Triangle, explaining the steps involved and the principles at work.

Understanding Pascal’s Triangle

Pascal’s Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. The structure of the triangle is such that the nth row (starting from 0) contains the coefficients for the binomial expansion of (a b)^n. To expand (a b)^n using Pascal’s Triangle, you need to find the corresponding row that has n 1 numbers.

Step 1: Identify the Appropriate Row

For the binomial (12y), the exponent is 3. Therefore, we need the 4th row in Pascal’s Triangle, which is the row with 4 numbers. Now, let's look at the 4th row in Pascal’s Triangle:

1331

Step 2: Apply the Binomial Expansion Rules

The general form of the binomial expansion is:

(a b)^n a^(n) na^(n-1)b (n(n-1)/2)a^(n-2)b^2 ... b^(n)

For our specific problem, we can break it down as follows:

12y^3 can be represented as a^0(b^3)12y^3, where a 1 and b 2y.

Step 3: Multiply with Pascal’s Coefficients

Using the numbers from the 4th row of Pascal’s Triangle, we multiply the terms accordingly:

11^3 31^2(2y) 31(2y)^2 1(2y)^3

Which simplifies to:

1 3(1)(2y) 3(1)(4y^2) 8y^3

1 6y 12y^2 8y^3

Step 4: Combine Like Terms

The terms 12y^2 do not affect the term we are interested in, which is 12y^3. Therefore, the final expanded form of 12y^3 using Pascal’s Triangle is:

1 6y 12y^3 8y^3

Conclusion

Using Pascal’s Triangle to expand binomials is a systematic and efficient method, especially useful in higher-level mathematics. By familiarizing yourself with the structure of Pascal’s Triangle and its coefficients, you can easily expand any binomial to the required power. Practice using different examples to solidify your understanding and apply this method effectively in your mathematical studies.