Problem 46 - Algebra 2

Question

The term 126\(c^{4} d^{5}\) appears in the expansion of \((c+d)^{n} .\) Find \(n\)

Step-by-Step Solution

Answer

The value of \( n \) is 9 in the expansion of \( (c+d)^n \) where the term 126\(c^{4}d^{5}\) appears.

Step 1: Understanding the binomial theorem

According to the binomial theorem, in the expansion of \( (a + b)^n \), the \( r^th \) term is given by \( nCr * a^{(n-r)} * b^r \), where nCr is the combination of 'n' items taken 'r' at a time. The powers on 'a' start with 'n' and decrease by 1 for each successive term, while the powers on 'b' start with 0 and increase by 1 for each successive term.

Step 2: Find the matching term

Seeing that the given term is 126\(c^{4}d^{5}\), it is clear that 'a' is assigned to 'c', 'b' is assigned to 'd'. The coefficient of the given term is 126. The power on \( c \) is 4, and the power on \( d \) is 5. Hence, we have to find the term \( nCr * c^{(n-r)} * d^r = 126c^{4}d^{5} \) from the expansion.

Step 3: Determine the value of n

Let's equate powers to find 'r' and 'n'. From \( c^{(n-r)} = c^{4} \), deduce that \( n-r = 4 \). From \( d^r = d^{5} \), deduce that \( r = 5 \). Substituting \( r = 5 \) in \( n-r = 4 \), find out that \( n = 9 \). 'n' is the power of binomial (c+d).

Step 4: Check the coefficient

To cross-verify, ensure that the value of n also satisfies the coefficient of the term. The given coefficient is 126. According to the binomial theorem, the coefficient should be \( 9C5 \), that is, choosing 5 terms from 9. Calculating that, it also gives 126, which matches the coefficient in the exercise.

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting and arrangements. In our exercise, it's linked with the binomial theorem. When expanding a binomial expression like \((a + b)^n\), combinatorics helps determine how many ways elements can be selected or arranged. In the binomial expansion, each term involves combinations, expressed as \(nCr\).

The notation \(nCr\) stands for "n choose r," representing the number of ways to select \(r\) items from \(n\). This concept is crucial as it dictates the coefficient in each term within the expansion. Given our exercise, when we needed to find a particular term, \(nCr\) helped determine both the arrangement of powers and the coefficient of that term in the binomial expansion.

Thinking of \(nCr\) as a way to **count possibilities** helps solve problems like our original exercise, where finding specific terms in polynomial expansions directly involves combinatorial calculations.

Polynomial Expansion

Polynomial expansion is the process where a binomial expression like \((a + b)^n\) is expanded into a full polynomial containing multiple terms. This occurs through the repeated application of the binomial theorem, which provides both structure and predictability in how terms appear, given specific exponents and coefficients.

In our example, \((c + d)^n\) was expanded to find a term matching \(126c^4d^5\). Each of these terms comes from systematically increasing and decreasing powers, structured by the binomial theorem. The powers of one variable decrease as you move through each term, while those of the other increase.

This method ensures that with practice, you can predict exactly how and what each term in a large polynomial will be by only knowing the original binomial expression.

Coefficient Calculation

Calculating coefficients in the binomial expansion is a crucial part of understanding polynomial forms. The coefficient of each term can be found using the combination formula \(nCr\), where \(r\) is the term's position within the expansion.

For example, in our exercise, the coefficient was given as 126 for the term \(c^4d^5\). By aligning this with the binomial formula, we see that the problem's solution involved verifying this result by ensuring that \(9C5\) also equals 126. Thus, it teaches that knowing the coefficient formula allows you not only to predict but verify terms within expansions.

Overall, understanding the role of coefficients means you can decipher any binomial expression's terms and more confidently solve related problems.