Question

In the expansion \((6+9 \mathrm{x})^{5}\) the coefficient of \(\mathrm{x}^{3}\) is (1) \(2^{2} \times 3^{x}\) (2) \(2^{4} \times 3^{7}\) (3) \(2^{3} \times 3^{x} \times 5\) (4) \(2^{4} \times 3^{7} \times 5\)

Short answer

Answer: The coefficient of \(x^3\) in the expansion is \(2^4 \times 3^7\).

Step-by-step solution

Write the binomial expansion formula for the given expression.

Using the binomial theorem, we can expand \((6+9x)^5\) as: \[((6+9x)^5 = \sum_{r=0}^5 C_5^r (6)^{5-r}(9x)^r\]

Find the term with \(x^3\) power by plugging in \(r=3\).

To find the term with \(x^3\), we need to find the term when \(r=3\): \[C_5^3 (6)^{5-3}(9x)^3\]

Calculate the binomial coefficient for \(r=3\).

Using the binomial coefficient formula, we can calculate \(C_5^3\) as: \[C_5^3 = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{120}{12} = 10\]

Substitute the binomial coefficient and simplify the term.

Now, substitute the binomial coefficient into the term and simplify: \[10 (6)^{2}(9x)^3 = 10 \times 36 \times 729x^3\]

Calculate the final coefficient of \(x^3\).

Multiply the constants in the term to find the coefficient of \(x^3\): \[10 \times 36 \times 729 = 2^4 \times 3^7\] Therefore, the coefficient of \(x^3\) in the expansion \((6+9x)^5\) is \(2^4 \times 3^7\), which corresponds to option (2).

Key concepts

Coefficients in Algebra

Understanding the role of coefficients in algebraic expressions is crucial for anyone studying algebra. Coefficients are the numeric parts that multiply the variables within an expression. In our case, we worked with the expression \( (6+9 \mathrm{x})^{5} \), aiming to find the coefficient of \( \mathrm{x}^{3} \). Identifying a specific coefficient involves expanding the expression and simplifying, as we did to determine the correct answer as \(2^{4} \times 3^{7}\), or option (2) in the exercise.

When dealing with coefficients, remember that they provide the scaling factor for each term, indicating how many times that term appears in the expansion. They are foundational in algebra because they allow us to understand and manipulate expressions effectively, paving the way towards solving equations. The coefficient's role is underscored when expressions become more complex, so mastering this concept early is beneficial for any student.

Binomial Expansion

The binomial expansion is a method of expanding expressions that are raised to a power, which in this case is the binomial \( (6+9 \mathrm{x})^{5} \). The binomial theorem provides a formula involving sums of powers, allowing us to expand the binomial without multiplying it out the long way. According to the theorem, the expression can be written as a sum of terms in the form \( C_n^r (a)^{n-r}(b)^r \), where \(a \) and \(b \) are the terms in the binomial, \(n \) is the exponent, and \( r \) is the term number. By plugging in the appropriate values for \( a \) and \( b \) and using combinatorics for the coefficients, we achieve the expanded form which includes the term we're interested in. In essence, binomial expansion transforms a compact binomial into a series of simpler algebraic terms.

Combinatorics

Combinatorics plays a pivotal role in the binomial expansion. It’s the mathematics of counting, and in our context, it applies to calculating the binomial coefficients \( C_n^r \), which appear in the expansion of a binomial. These coefficients are also known as combination numbers because they represent the different ways to choose \( r \) elements from a set of \( n \) without regard to the order.

For instance, the term \( C_5^3 \) calculated in our solution represents the number of ways to choose 3 items from a set of 5 distinct items, which is crucial to finding the exact term containing \( x^3 \). This interplay between combinatory calculations and algebraic principles underscores not just the solution to our exercise but reveals the broader importance of combinatorics in elaborating the structure of binomial expansion.

Mathematical Induction

Mathematical induction is a technique for proving statements or formulas that assert something about an infinite set of natural numbers. Though not explicitly used in solving our exercise, it's a fundamental concept often utilized to prove the validity of the binomial theorem itself. Imagine wanting to prove that the binomial theorem holds true for all positive exponents. Mathematical induction allows us to first verify the base case (usually when the exponent is 1). Then we assume the statement is true for some natural number \( k \) and prove it for \( k+1 \).

This step-by-step approach aligns perfectly with the incremental nature of comprehension that we aim for in education. Inductive reasoning like this is indispensable in mathematics, because it builds our understanding from specific cases to general truths – an endeavor not unlike the way we gradually unpack the components of complex algebraic expressions like binomial expansions.