Problem 45 - A Graphical Approach to College Algebra

Question

Find the middle term of \(\left(3 x^{7}+2 y^{3}\right)^{8}\).

Step-by-Step Solution

Answer

The middle term is \(90720x^{28}y^{12}\).

Step 1: Identify the Binomial Theorem

The binomial theorem helps us expand expressions of the form \((a + b)^n\). It states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is a binomial coefficient.

Step 2: Determine the Values for the Binomial Expansion

In this problem, identify \(a = 3x^7\), \(b = 2y^3\), and \(n = 8\). We need to find the middle term in this expansion. Since \(n = 8\), the expansion will have \(n+1 = 9\) terms. The middle term for an even \(n\) is the \(\frac{n}{2} + 1\)th term, which in this case is the 5th term.

Step 3: Calculate the 5th Term

The \(k\)th term in the binomial expansion is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). Therefore, the 5th term \(T_5\) corresponds to \(k = 4\). Thus, calculate \(T_5 = \binom{8}{4} (3x^7)^{8-4} (2y^3)^4\).

Step 4: Evaluate the Binomial Coefficient

Calculate \(\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\).

Step 5: Simplify the Powers of each Term

Calculate \((3x^7)^4 = 3^4 \cdot (x^7)^4 = 81x^{28}\) and \((2y^3)^4 = 2^4 \cdot (y^3)^4 = 16y^{12}\).

Step 6: Combine all the Parts of the 5th Term

Multiply all parts of the term together: \(T_5 = 70 \cdot 81x^{28} \cdot 16y^{12}\). Simplify this to get \(T_5 = 90720x^{28}y^{12}\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful method used to expand expressions that are raised to a power, specifically of the form \((a + b)^n\). This concept is rooted in the Binomial Theorem, which provides a structured formula for achieving such expansions.
The formula for binomial expansion is expressed as:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Here, \(a\) and \(b\) can be any terms, and \(n\) is the exponent to which the expression is raised.
This expansion results in \(n+1\) terms, each representing a distinct combination of the powers of \(a\) and \(b\). For instance, in the expression \((3x^7 + 2y^3)^8\), \((a, b) = (3x^7, 2y^3)\) and \(n = 8\). The expansion will have nine terms in total. The binomial expansion unfolds systematically to reveal the polynomial terms consisting of various powers of \(a\) and \(b\).
Each term in the expansion is accompanied by a binomial coefficient, which defines how many different ways that particular arrangement of powers can occur.

Binomial Coefficient

The binomial coefficient is a key component of the binomial expansion. These are specific numbers that multiply each term in the expanded form and are represented by \(\binom{n}{k}\), which is pronounced as "n choose k".
Mathematically, the binomial coefficient \(\binom{n}{k}\) can be calculated using the formula:

• \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

Where \(!\) denotes factorial, which is the product of all positive integers up to that number.
For example, if you have \(n = 8\) and you want the 5th term (which corresponds to \(k = 4\) because terms start at \(k = 0\)), you calculate:

• \[\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\]

This coefficient, \(70\), plays a crucial role in determining the weight or significance of that particular term in the expansion. It's crucial to perform these calculations properly, as they directly impact the expanded form of the binomial expression.

Polynomial Terms

In the binomial expansion, each sequence of multiplied components forms what is called a polynomial term. When expanding \((3x^7 + 2y^3)^8\), polynomial terms are generated, combining coefficients and powers.
Each polynomial term in the expansion can be described by the general term formula:

• \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\)

Let's take a closer look at a specific term in our example: the 5th term, which arises when \(k = 4\). We calculate it as follows:

• The coefficient \(\binom{8}{4} = 70\)
• The power \((3x^7)^4 = 81x^{28}\)
• The power \((2y^3)^4 = 16y^{12}\)

Combining these, we form the 5th polynomial term:

• \(T_5 = 70 \times 81x^{28} \times 16y^{12}\)
• This simplifies to \(90720x^{28}y^{12}\)

Each polynomial term provides specific information about the expanded form, reflecting the distinct interaction between the bases and their respective powers.