Problem 10 - Algebra 2

Question

Find the indicated term of each expansion. fifth term of \((2 a+3 b)^{10}\)

Step-by-Step Solution

Answer

The fifth term of the expansion is \( 1088640a^6b^4 \).

Step 1: Recognize the formula

The expansion of \(a+b\)^n follows the binomial theorem, which states: \[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k, \] where \ \binom{n}{k} \ is a binomial coefficient.

Step 2: Identify variables

In this problem, \(a = 2a\), \(b = 3b\), and \(n = 10\). The fifth term corresponds to \(k = 4\) in the general expansion formula.

Step 3: Calculate binomial coefficient

For the fifth term, use \ \binom{10}{4} \ to find the binomial coefficient:\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \].

Step 4: Apply binomial formula to find the term

The fifth term is given by substituting \(n = 10\) and \(k = 4\) into the binomial expansion formula: \[ \binom{10}{4} (2a)^{10-4} (3b)^4 = 210 \times (2a)^6 \times (3b)^4 \].

Step 5: Simplify the expression

Compute the powers and multiply together: - \( (2a)^6 = 64a^6 \)- \( (3b)^4 = 81b^4 \)Multiply these with the binomial coefficient:\[ 210 \times 64a^6 \times 81b^4 = 1088640a^6b^4 \].

Key Concepts

Binomial Coefficient

The binomial coefficient is a key component in the Binomial Theorem. It is often represented in mathematics as \( \binom{n}{k} \), pronounced as "n choose k." This notation signifies the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to order. It is calculated using the formula:

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

The exclamation marks denote factorials, which mean the product of all positive integers up to that number. For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
In our exercise, we calculate \( \binom{10}{4} \) to be 210. This result tells us that there are 210 different ways to choose 4 items from a set of 10. In the context of our polynomial expansion, this number serves as the coefficient of the specific term, indicating its importance in the expression.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial raised to a power in an expanded form using the Binomial Theorem. The theorem provides a systematic way of expanding expressions like \((a+b)^n\) into a sum of terms, each involving powers of \(a\) and \(b\).
In our example, we expanded \((2a + 3b)^{10}\). To derive each term, we follow the binomial expansion formula:

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

Each term in the expanded polynomial consists of:

• a binomial coefficient \( \binom{n}{k} \),
• the variable 'a' raised to a power \( n-k \),
• and the variable 'b' raised to a power \( k \).

The fifth term is particularly obtained by considering \(k=4\), leading to a specific coefficient and powers for the variables after application and simplification. This structured approach allows us to systematically find any term in the expansion.

Algebraic Expressions

Algebraic expressions are made up of variables and constants combined using operations such as addition, subtraction, multiplication, and division. In our exercise, the expression \((2a + 3b)^{10}\) is an example of a compound algebraic expression consisting of two terms in a binomial format.
When working with algebraic expressions, understanding the roles of coefficients and exponents is crucial. The coefficients, such as the resulting 210 in our binomial coefficient calculation, multiply the variables' terms to provide the magnitude of each component in the expression.
Variables like \(a\) and \(b\) represent quantities that can change, and when raised to powers, they indicate repeated multiplication. For instance, in the expression \((2a)^6\), the 6 means \(2a\) is multiplied by itself six times, resulting in \(64a^6\). Simplifying these expressions helps in understanding complex polynomial expansions by providing concrete values for computations, aligning with the calculation seen in the fifth term of our original binomial expansion.