Can You Use Pascal’s Triangle to Expand the Binomial (2x)^3(3y)^15?

Do you find yourself needing to expand binomial expressions without relying on complex formulas? If so, learning to use Pascal’s Triangle can be a straightforward and effective method. This guide will walk you through the process, making it accessible and understandable, even if you are just starting with this concept.

Introduction to Pascal’s Triangle

Pascal’s Triangle is a powerful tool in mathematics, rooted in its ability to generate binomial coefficients. Each row in the triangle corresponds to the coefficients of the expanded form of ((a b)^n), where (n) is a non-negative integer and (a, b) are variables or constants. The triangle is constructed in such a way that each element is the sum of the two elements directly above it in the previous row.

Understanding Binomial Expansion

Binomial expansion involves breaking down the expression ((a b)^n) into a sum of terms, where each term is a product of a coefficient and powers of (a) and (b). The coefficients are derived from Pascal's Triangle, and each coefficient corresponds to a specific position in the expansion. For example, the expansion of ((a b)^4) would be:

[(a b)^4 a^4 4a^3b 6a^2b^2 4ab^3 b^4]

The coefficients here are 1, 4, 6, 4, 1, which form a row in Pascal's Triangle.

Applying Pascal’s Triangle to Binomial Expansion

Let’s consider the given binomial expression ((2x)^3(3y)^{15}). To expand this using Pascal’s Triangle, first, we need to identify the binomial coefficient using the triangle. For simplicity, let's break it down step by step:

Step 1: Simplify the Expression

First, we can simplify the expression by distributing the powers over the constants.

[ (2x)^3(3y)^{15} 2^3 x^3 cdot 3^{15} y^{15} ]

Now, let's focus on the binomial part. In this case, the binomial part is simply ((x y)^{18}).

Step 2: Construct Pascal’s Triangle

To do the expansion, we need the coefficients from the 18-th row of Pascal’s Triangle. Here is how the row looks:

1 18 153 816 3060 7381 12376 12376 7381 3060 816 153 18 1

Step 3: Apply the Coefficients to the Binomial Expansion

Now, we apply these coefficients to the corresponding terms in the expansion of ((x y)^{18}):

[(x y)^{18} x^{18} 18x^{17}y 153x^{16}y^2 816x^{15}y^3 306^{14}y^4 7381x^{13}y^5 12376x^{12}y^6 12376x^{11}y^7 7381x^{10}y^8 306^9y^9 816x^8y^{10} 153x^7y^{11} 18x^6y^{12} y^{18}]

Step 4: Multiply by the Constants

Finally, we multiply the entire expansion by (2^3 cdot 3^{15}):

[(2x)^3(3y)^{15} 8 cdot 14348907 cdot (x^{18} 18x^{17}y 153x^{16}y^2 816x^{15}y^3 306^{14}y^4 7381x^{13}y^5 12376x^{12}y^6 12376x^{11}y^7 7381x^{10}y^8 306^9y^9 816x^8y^{10} 153x^7y^{11} 18x^6y^{12} y^{18})]

Which simplifies to:

[ (2x)^3(3y)^{15} 11477395456x^{18} 20669211816x^{17}y 184990664064x^{16}y^2 1067339961632x^{15}y^3 4268049616192x^{14}y^4 13188864843584x^{13}y^5 31277113885696x^{12}y^6 31277113885696x^{11}y^7 21741474909952x^{10}y^8 12663554886848x^9y^9 5262849779248x^8y^{10} 1688655437176x^7y^{11} 401489198496x^6y^{12} 14348907y^{18})]

Conclusion

Using Pascal’s Triangle to expand binomials can greatly simplify your work and ensure accuracy. With practice, you can quickly apply the pattern and produce the correct results.

Tips for Practicing

Start with simpler binomials to get familiar with the process. Practice with different values for (a) and (b). Try expanding binomials with higher exponents to further solidify your understanding.

Now, you are ready to use Pascal's Triangle to tackle any binomial expansions you encounter! Remember, the key is to understand the pattern and practice consistently.