Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (3 a+b)^{20} $$

Step-by-Step Solution

Answer

The first three terms are \(3^{20} a^{20}, 20 \times 3^{19} a^{19} b, 190 \times 3^{18} a^{18} b^2\).

Step 1: Identify the General Form of the Binomial Theorem

The Binomial Theorem states that for any non-negative integer \( n \), \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). This theorem helps us expand expressions like \((x+y)^n\) into a series.

Step 2: Assign Values to Variables

In the expression \((3a + b)^{20}\), identify \(x = 3a\), \(y = b\), and \(n = 20\). These values will be used in the Binomial Theorem.

Step 3: Calculate the First Term

The first term of the expansion corresponds to \(k = 0\). According to the formula, the first term is \(\binom{20}{0} (3a)^{20} b^0\). Since \(\binom{20}{0} = 1\) and \(b^0 = 1\), the result is \( (3a)^{20} = 3^{20} a^{20}\).

Step 4: Calculate the Second Term

For the second term, \(k = 1\). Using the formula, the second term is \(\binom{20}{1} (3a)^{19} b^1 = 20 \times (3a)^{19} b\). Simplifying further gives \(20 \times 3^{19} a^{19} b\).

Step 5: Calculate the Third Term

For the third term, \(k = 2\). Apply the formula: \(\binom{20}{2} (3a)^{18} b^2 = 190 \times (3a)^{18} b^2\). This simplifies to \(190 \times 3^{18} a^{18} b^2\).

Step 6: Write the First Three Terms

Putting it all together, the first three terms of the expansion are: \(3^{20} a^{20}, 20 \times 3^{19} a^{19} b, 190 \times 3^{18} a^{18} b^2\).

Key Concepts

Binomial Expansion

The Binomial Expansion is a powerful tool in mathematics for expanding expressions of the form \((x + y)^n\) into a series. It relies on the Binomial Theorem, which states:

• If you have a binomial (a two-term expression), such as \((x + y)\), and you raise it to a power \(n\), you can expand it into a sum of terms using a specific formula.
• This formula involves binomial coefficients, given by \(\binom{n}{k}\), which are calculated using combinations.
• The general form of each term is \(\binom{n}{k} x^{n-k} y^k\), where \(k\) ranges from 0 to \(n\).

To see it in action, let's consider the example \((3a + b)^{20}\). By assigning \(x = 3a\), \(y = b\), and \(n = 20\), we can use the theorem to expand the expression term-by-term. This expansion starts with high powers of \(a\) and then incrementally powers up \(b\) while decreasing powers of \(a\). Binomial Expansion is not just a mathematical exercise; it helps solve problems in probability, series, and even calculus!

Combinatorics

Combinatorics is crucial to understanding the Binomial Expansion. It revolves around the study of counting, arrangements, and combinations of objects. In the context of the Binomial Theorem, combinatorics introduces us to binomial coefficients \(\binom{n}{k}\).

• A binomial coefficient \(\binom{n}{k}\) tells us how many ways we can choose \(k\) elements from a total of \(n\) elements without considering the order.
• The formula for a binomial coefficient is \(\binom{n}{k} = \frac{n!}{k! (n-k)!}\), where \(!\) indicates factorial.

In the expansion of \((3a + b)^{20}\), combinatorics helps compute each term's coefficient, beginning with \(\binom{20}{0}\), \(\binom{20}{1}\), and \(\binom{20}{2}\) for the first few terms. Practically, these coefficients can grow large quickly, but combinatorics provides formulas and tools to calculate them efficiently. Through combinatorics, we gain the power to handle complex counting problems, especially in probabilistic scenarios or when dealing with polynomial expansions.

Polynomial Expressions

Polynomial expressions are algebraic expressions that involve sums of powers of variables. Each term in a polynomial includes a coefficient and a power of a variable. When we think about polynomial expressions, a binomial expansion can be seen as a specific way of generating polynomials from a binomial raised to a power.

• Each term in the expansion of a binomial polynomial is itself a polynomial term, characterized by its distinct combination of variable powers and a coefficient.
• This is evident in the expression of \((3a + b)^{20}\), where each resulting term like \(3^{20}a^{20}\) or \(20 \times 3^{19}a^{19}b\) is a polynomial term.
• Polynomial expressions are versatile and occur in a multitude of mathematical problems and real-life scenarios.

The expansion transforms a simple-looking expression into a detailed polynomial, helping provide solutions or approximations in various fields of science and engineering. Also, it showcases the inherent beauty and symmetry that such expansions naturally exhibit.