Problem 23 - College Algebra

Question

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (a+b)^{17} $$

Step-by-Step Solution

Answer

The first three terms are \(a^{17} + 17a^{16}b + 136a^{15}b^2.\)

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a formula for expanding binomials raised to any positive integer power. It is given by: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) represents the binomial coefficients.

Step 2: Identify Values

In the expression \((a+b)^{17}\), the values are \(n=17\), \(a=a\), and \(b=b\). You will use these to calculate the first three terms using the binomial expansion formula.

Step 3: Calculate the First Term

The first term occurs when \(k=0\). Substitute into the formula: \[\binom{17}{0} a^{17-0} b^0 = 1 \cdot a^{17} \cdot 1 = a^{17}.\]

Step 4: Calculate the Second Term

The second term occurs when \(k=1\). Substitute into the formula: \[\binom{17}{1} a^{17-1} b^1 = 17 \cdot a^{16} \cdot b = 17a^{16}b.\]

Step 5: Calculate the Third Term

The third term occurs when \(k=2\). Substitute into the formula: \[\binom{17}{2} a^{17-2} b^2 = \frac{17 \cdot 16}{2} a^{15} b^2 = 136a^{15}b^2.\]

Step 6: Write Down the First Three Terms

Combine the results from Steps 3, 4, and 5 to write the first three terms of the expansion: \[a^{17} + 17a^{16}b + 136a^{15}b^2.\]

Key Concepts

Binomial Coefficients

Binomial coefficients are an essential part of the Binomial Theorem. They represent the coefficients in the expanded form of a binomial raised to a power. In the formula \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), the term \(\binom{n}{k}\) is known as a binomial coefficient.

These coefficients are calculated using combinations, where \(\binom{n}{k}\) is the number of ways to choose \(k\) elements from a set of \(n\) elements. The formula for calculating binomial coefficients is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, \(!\) represents the factorial, which is the product of all positive integers up to that number.

For example:

• \(\binom{17}{0} = 1\) since choosing 0 items from 17 always has one way.
• \(\binom{17}{1} = 17\) because there are 17 ways to choose one item out of 17.
• \(\binom{17}{2} = 136\) calculated as \(\frac{17 \cdot 16}{2}\) because there are 136 ways to choose 2 items from 17.

These coefficients make it possible to expand any binomial raised to a power efficiently.

Binomial Expansion

The binomial expansion is a technique used to expand expressions that are raised to a power, such as \((a+b)^{17}\). According to the Binomial Theorem, the binomial expression can be expanded into a series of terms using binomial coefficients. This involves calculating the individual terms of the expansion.

For the expression \((a+b)^{17}\), the expansion is derived as follows:

• The first term is found by setting \(k=0\), resulting in \(a^{17}\).
• The second term is obtained by setting \(k=1\), which gives \(17a^{16}b\).
• The third term is calculated by setting \(k=2\), resulting in \(136a^{15}b^2\).

By following this pattern, one can compute all terms in the series until \(k=17\). Each term consists of a binomial coefficient multiplied by powers of \(a\) and \(b\), with the sum of the powers always equaling the original exponent, in this case, 17. The pattern ensures that all possibilities of distributing power between \(a\) and \(b\) are covered.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is used to express mathematical concepts in a generalized form. The Binomial Theorem is deeply rooted in algebra as it involves the manipulation and understanding of polynomials and their expansions.

In a binomial expansion like \((a+b)^{17}\), algebra helps translate the problem into a series of steps that can be solved systematically. Each term in the expansion uses algebraic expressions and operations:

• Variables \(a\) and \(b\), which are manipulated according to the rules of exponents.
• Operations such as addition and multiplication, which combine these variables with binomial coefficients.

Through algebra, we not only find the expanded form of a binomial expression but also gain a deeper understanding of how these expansions relate to combinations and the principles of exponentiation. This kind of algebraic thinking is foundational across many areas of mathematics, making the study of binomial expansions an important topic for developing mathematical skills.