Problem 47 - A Graphical Approach to College Algebra

Question

Find the value of \(n\) for which the coefficients of the fifth and eighth terms in the expansion of \((x+y)^{n}\) are the same.

Step-by-Step Solution

Answer

The value of \(n\) is 11.

Step 1: Identify the General Term Formula

The general term in the expansion of \((x+y)^n\) is given by \(T_{k+1} = \binom{n}{k}x^{n-k}y^k\). In this exercise, we are asked to compare the coefficients of the 5th and 8th terms, so we need the 4th and 7th powers of \(y\). Therefore, the fifth term corresponds to \(k=4\) and the eighth term corresponds to \(k=7\).

Step 2: Set Up the Equation

The coefficients of the fifth and eighth terms must be equal. This implies that: \[ \binom{n}{4} = \binom{n}{7} \] This equation represents the equality of the binomial coefficients of the 5th and 8th terms.

Step 3: Use Binomial Coefficient Property

Use the property \(\binom{n}{k} = \binom{n}{n-k}\). Thus, the equation \(\binom{n}{4} = \binom{n}{7}\) can be simplified using \(\binom{n}{7} = \binom{n}{n-7}\). So, we can equate \(4\) to \(n-7\).

Step 4: Solve for n

From the equation obtained in step 3, we have: \[ 4 = n - 7 \] Solve this equation to find \(n\).

Step 5: Simplify the Equation

Add 7 to both sides of the equation: \[ 4 + 7 = n \] \[ n = 11 \] Thus, the value of \(n\) is 11.

Key Concepts

Coefficient Analysis

Coefficient analysis in binomial expansions is crucial for finding relationships between different terms. When you look at an expansion such as \( (x+y)^n \), you focus on specific terms to pinpoint their coefficients. For example, each term in the binomial expansion is represented by \( T_{k+1} = \binom{n}{k} x^{n-k} y^k \). This formula aids in identifying the coefficient for any given term, where \( \binom{n}{k} \) is the binomial coefficient.

• The binomial coefficient, \( \binom{n}{k} \), essentially tells you how many ways to choose \( k \) items out of \( n \).
• For instance, in our case, we concentrated on the 5th term with \( k=4 \) and the 8th term with \( k=7 \).

By finding when these coefficients are equivalent, unknowns in the expansion such as the exponent \( n \) become solvable. This analysis involves understanding the structure of the expansion to find specific values or solve for variable expressions.

Binomial Coefficient Equality

The idea of binomial coefficient equality arises naturally when two terms in a polynomial expansion need to have the same coefficient. If you are given a situation like this, it means you'll likely be working with the property:\[ \binom{n}{k} = \binom{n}{n-k} \]This property is symmetrical, meaning it doesn't matter from which angle you look at the combination. The beauty here is that it allows simplification by directly setting equal coefficients against each other.

• In the example problem, we established \( \binom{n}{4} = \binom{n}{7} \).
• This relationship used the symmetry property to conclude that the terms positioned at complementary indexes have equal fillings.

To gain solutions, this property often simplifies problems, leading to straightforward arithmetic equations that unravel the unknowns elegantly.
This exploration provides a bridge between theoretical mathematics and tangible solutions, revealing how such equalities can turn complex scenarios into solvable puzzles.

Polynomial Expansion

Polynomial expansion through the binomial theorem transforms expressions like \((x+y)^n\) into a sum of multiple terms. Each term displays a combination of both powers of \(x\) and \(y\), comprising simple polynomial chunks that add together to reassemble the original higher power expression.In_context, consider each term is represented by the structure\( T_{k+1} = \binom{n}{k} x^{n-k} y^k \), where \( n \) is the overall power, \( k \) identifies the specific term, and the coefficients depict how power is distributed between \(x\) and \(y\).

• Expanding a polynomial like \((x+y)^n\) involves determining all these terms to fully express it as a polynomial of degree \( n \).
• The summation covers all values from \( k=0 \) to \( n \), delivering a meticulous formulation for any given exponent.

By systematically expanding the polynomial, the process reveals intricate patterns of coefficients and powers that are fundamental for algebraic transformation, making this method highly valuable for structuring algebraic expressions. The expansion holds a tale of symmetry and pattern, essential for dissecting larger polynomial structures to their core.