Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Expand the binomial by using Pascal's Triangle to determine the coefficients. \((2 v+3)^{6}\)

Step-by-Step Solution

Answer

The expanded form of \((2v+3)^6\) using Pascal's Triangle is \(64v^6 + 576v^5 + 21600v^4 + 86400v^3 + 48600v^2 + 2916v + 729\).

Step 1: Construct Pascal's Triangle

First, Pascal's Triangle needs to be constructed up to the 6th row. Starting from the top, each number in the triangle is the sum of the two numbers directly above it. The 6th row of Pascal's Triangle is: 1, 6, 15, 20, 15, 6, 1.

Step 2: Separate the Binomial

The binomial given is \((2v+3)^6\). Here, \(a\) is \(2v\) and \(b\) is \(3\). Now consider each term of the expanded version of the binomial: it will be of the form \(\binom{6}{r}a^{6-r}b^r\), where \(\binom{6}{r}\) is the r-th element of the 6th row of Pascal's Triangle.

Step 3: Apply Binomial Theorem

Applying the Binomial Theorem, each term is calculated as follow:\n\n Term 1: \(\binom{6}{0}(2v)^{6-0}3^0 = 1 * 64v^6 * 1 = 64v^6\)\n\n Term 2: \(\binom{6}{1}(2v)^{6-1}3^1 = 6 * 32v^5 * 3 = 576v^5\)\n\n Term 3: \(\binom{6}{2}(2v)^{6-2}3^2 = 15 * 16v^4 * 9 = 21600v^4\)\n\n Term 4: \(\binom{6}{3}(2v)^{6-3}3^3 = 20 * 8v^3 * 27 = 86400v^3\)\n\n Term 5: \(\binom{6}{4}(2v)^{6-4}3^4 = 15 * 4v^2 * 81 = 48600v^2\)\n\n Term 6: \(\binom{6}{5}(2v)^{6-5}3^5 = 6 * 2v* 243 = 2916v\)\n\n Term 7: \(\binom{6}{6}(2v)^{6-6}3^6 = 1 * 1 * 729 = 729\)

Step 4: Combine the Terms

Now, all of the terms are combined into the expanded version of the given binomial: \n\n \(64v^6 + 576v^5 + 21600v^4 + 86400v^3 + 48600v^2 + 2916v + 729\)

Key Concepts

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers which shows coefficients in the binomial expansion. It starts with a 1 at the top and each row corresponds to coefficients for an increasing power of a binomial expression. For example, the first row is for \((a+b)^0\), the second for \((a+b)^1\), and so on.
The process of constructing Pascal's Triangle involves each number being the sum of the two numbers directly above it, forming a symmetrical pattern. For instance, to build the 6th row, which is needed for this problem, you start with:

• 1,
• then 1 + 5 = 6,
• followed by 5 + 10 = 15,
• next comes 10 + 10 = 20,
• then 10 + 5 = 15,
• and finally 5 + 1 = 6,
• end with another 1.

This results in the 6th row being: 1, 6, 15, 20, 15, 6, 1.

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form \((a + b)^n\) into a sum of terms. It states that:\[(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r}b^r\]Each term in the binomial expansion corresponds to a combination of powers of \(a\) and \(b\) multiplied by a coefficient from Pascal's Triangle.
For the given problem, \((2v+3)^6\), the theorem outlines how to calculate each term:

• Powers of 2v and 3: The powers of 2v and 3 in a term must add up to 6, the power of the binomial expression.
• Selection of Coefficients: Use coefficients from the 6th row of Pascal's Triangle to contribute the multiplicative factor for each term.

Applying this theorem simplifies expanding binomials effectively by following a systematic approach.

Polynomial Coefficients

Polynomial coefficients are the numbers that scale the terms in a polynomial, providing the constant factor that multiplies the variables raised to their respective powers. In the context of binomial expansion:
These coefficients come from Pascal's Triangle and are crucial for determining each term in an expansion. Simply choosing these coefficients correctly ensures the accuracy of your expanded polynomial.
For \((2v+3)^6\), the polynomial coefficients from the 6th row of Pascal's Triangle are used as shown:

• 1 scales \(64v^6\),
• 6 scales \(576v^5\),
• 15 scales \(21600v^4\),
• 20 scales \(86400v^3\),
• 15 scales \(48600v^2\),
• 6 scales \(2916v\),
• 1 scales \(729\).

Polynomial coefficients work hand-in-hand with the variable components to give the final expanded form of the polynomial expression: \(64v^6 + 576v^5 + 21600v^4 + 86400v^3 + 48600v^2 + 2916v + 729\).