Problem 93 - Algebra for College Students

Question

We have used the following two multiplication patterns. $$ \begin{aligned} &(a+b)^{2}=a^{2}+2 a b+b^{2} \\ &(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \end{aligned} $$ By multiplying, we can extend these patterns as follows: $$ \begin{aligned} &(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ &(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a^{4}+b^{5} \end{aligned} $$ On the basis of these results, see if you can determine a pattern that will enable you to complete each of the following without using the long- multiplication process. (a) \((a+b)^{6}\) (b) \((a+b)^{7}\) (c) \((a+b)^{8}\) (d) \((a+b)^{9}\)

Step-by-Step Solution

Answer

Use the binomial theorem to expand \((a+b)^n\), utilizing Pascal's triangle for coefficients.

Step 1: Identify the Pattern

Notice that the expansion of \((a+b)^n\) forms a sequence based on the binomial theorem, where the coefficients are known as binomial coefficients. They correspond to the entries in Pascal's triangle.

Step 2: Use Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient and can be found using Pascal's triangle or the formula \(\binom{n}{k}=\frac{n!}{k!(n-k)!}\).

Step 3: Calculate \\(a+b\\)^6

Use the binomial theorem: \((a+b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6\). These coefficients are derived from the 6th row of Pascal's triangle: 1, 6, 15, 20, 15, 6, 1.

Step 4: Calculate \\(a+b\\)^7

The expansion is \((a+b)^7 = a^7 + 7a^6b + 21a^5b^2 + 35a^4b^3 + 35a^3b^4 + 21a^2b^5 + 7ab^6 + b^7\). These coefficients correspond to the 7th row of Pascal's triangle: 1, 7, 21, 35, 35, 21, 7, 1.

Step 5: Calculate \\(a+b\\)^8

Expand using the binomial theorem: \((a+b)^8 = a^8 + 8a^7b + 28a^6b^2 + 56a^5b^3 + 70a^4b^4 + 56a^3b^5 + 28a^2b^6 + 8ab^7 + b^8\). The coefficients are from the 8th row of Pascal's triangle: 1, 8, 28, 56, 70, 56, 28, 8, 1.

Step 6: Calculate \\(a+b\\)^9

Apply the binomial theorem: \((a+b)^9 = a^9 + 9a^8b + 36a^7b^2 + 84a^6b^3 + 126a^5b^4 + 126a^4b^5 + 84a^3b^6 + 36a^2b^7 + 9ab^8 + b^9\). These coefficients align with the 9th row of Pascal's triangle: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1.

Key Concepts

Pascal's Triangle

Pascal's Triangle is a fascinating mathematical concept that reveals the coefficients for the expansion of binomial expressions like \((a+b)^n\). This triangle starts with a single 1 at the top, and as you move down each row, it continues to grow wider. Each row corresponds to the powers of \((a+b)^n\), where \(n\) is the row number starting from zero.

Here's how it works:

• The first and last number of each row is always 1.
• Each number inside the triangle is the sum of the two numbers directly above it in the previous row.

For example, the third row of Pascal’s Triangle is 1, 2, 1. If you expand \((a+b)^2\), the coefficients match exactly to the numbers in this row: 1, 2, and 1.

Pascal's Triangle not only provides the coefficients for binomial expansions but also holds magical patterns and numbers like fibonacci sequences that entice mathematicians everywhere!

Binomial Coefficients

Binomial Coefficients are a critical aspect of the Binomial Theorem, and they determine the weights or factors each term in a binomial expansion receives. You encounter these when looking at terms in an expression like \((a+b)^n\). Each term in such a polynomial is accompanied by a binomial coefficient.

These coefficients can be calculated using either the factorial formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]or by referencing Pascal's Triangle, which visually represents these numbers.

Using the factorial method:

• \(n!\) (n factorial) means multiplying all whole numbers from 1 to \(n\).
• \(k\) represents the term's position in the expansion, starting at 0.
• Calculate \(n!\), \(k!\), and \((n-k)!\) and use them in the formula.

This calculation gives the exact number needed as the binomial coefficient for any term, showing how many times each product of \(a\) and \(b\) occurs in the expansion.

Algebraic Expansion

Algebraic Expansion simply refers to the process of expanding an expression raised to a power, like transforming \((a+b)^n\) into a polynomial. It involves multiplying the expression by itself \(n\) times, but there's no need to manually do this when the Binomial Theorem is at hand.

The Binomial Theorem offers a formulaic approach that simplifies this task by using binomial coefficients:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Each term in the expanded form of a binomial expression is structured as:

• \(\binom{n}{k}\): The binomial coefficient, indicating how many of each term exist.
• \(a^{n-k}\): The variable \(a\) is raised to the power of \((n-k)\).
• \(b^k\): The variable \(b\) is raised to the power of \(k\).

Through this methodical expansion, you can quickly and efficiently break down expressions like \((a+b)^6\), precisely pinpointing every term and its respective coefficient without laborious multiplication efforts.