Problem 36 - Algebra 2

Question

Find the indicated term of each expansion. fourth term of \((2 x+3 y)^{9}\)

Step-by-Step Solution

Answer

The fourth term is \(145152x^6y^3\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem is given by the formula \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\], where \(\binom{n}{k}\) is the binomial coefficient. In our case, \(a = 2x\), \(b = 3y\), and \(n = 9\).

Step 2: Determine the Index for the Fourth Term

According to the Binomial Theorem, the term in the expansion is labeled with index \(k+1\). Since we need the fourth term, the corresponding \(k\) is \(4 - 1 = 3\).

Step 3: Find the Binomial Coefficient

For the fourth term, calculate the binomial coefficient \(\binom{9}{3}\), which is computed as follows: \[\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84.\]

Step 4: Calculate Powers of Each Term

The generic form of the term is given by \[\binom{9}{3} (2x)^{9-3} (3y)^3.\] Calculate the powers: - \((2x)^{6} = 64x^6\),- \((3y)^3 = 27y^3\).

Step 5: Compute the Fourth Term

Multiply the binomial coefficient with the powers calculated: \[84 \times 64x^6 \times 27y^3 = 145152x^6y^3.\]

Step 6: Formulate the Final Expression

The fourth term of the expansion is \[145152x^6y^3.\]

Key Concepts

Binomial Expansion

The concept of binomial expansion is rooted in the Binomial Theorem, a handy tool in algebra for expanding expressions that involve powers of binomials. This theorem provides a systematic way to expand powers of the form \( (a+b)^n \). The formula for a binomial expansion is:

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

This means that when you expand \( (a+b)^n \), you end up with a polynomial consisting of multiple terms. Each term in this expansion comes from a combination of coefficients and involved powers of \( a \) and \( b \). The expansion breaks down a complex expression into more manageable parts, which is particularly useful for solving algebraic problems or simplifying expressions in various fields, like calculus or probability theory.

When tackling a problem like the given exercise, understanding this expansion is crucial. We are essentially breaking down and calculating apurely algebraic expression into simpler terms using the powers of given components.

Binomial Coefficient

A critical part of any binomial expansion is the calculation of binomial coefficients, denoted by \( \binom{n}{k} \). These coefficients are essential, as they determine the weighting of each term in the expansion. The formula for finding a binomial coefficient is:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
Here, \( n! \) signifies the factorial of \( n \), representing the product of all integers from 1 to \( n \). Calculating these coefficients accurately is crucial, as they tell us how much each individual term should "contribute" to the complete expansion.
In the provided exercise, we find the fourth term of \( (2x+3y)^9 \) by setting \( k = 3 \) (because the index \( k \) is one less than the term's position). The calculated binomial coefficient, \( \binom{9}{3} = 84 \), helps us weight the power terms \((2x)^6\) and \((3y)^3\) correctly in the expansion.
Getting these coefficients right is essential, as every term in a binomial expansion has its own unique coefficient that helps define its place in the polynomial.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial power in expanded form, comprising individual monomials. Each monomial in the expansion is obtained by applying the binomial theorem, specifically through calculating respective powers and multiplying them by their binomial coefficients.

In practice, the process involves calculating and organizing terms based on degrees of each variable, resulting in a fully expanded polynomial that can be easier analyzed or used further in algebraic operations.

For example, in the original exercise, the goal was to find a specific term within the polynomial expansion of \( (2x + 3y)^9 \). Using the calculated binomial coefficient \( \binom{9}{3} \), we determine:

• The power of \( 2x \), computed as \( (2x)^6 = 64x^6 \)
• The power of \( 3y \), computed as \( (3y)^3 = 27y^3 \)

With these, we form the desired term \( 145152x^6y^3 \), highlighting how polynomial expansion helps us break down a large power into manageable algebraic terms. This method is instrumental in fields ranging from engineering to physics, where precise polynomial calculations are often necessary.