Question

Use the given value of \(n\) to find the coefficient of \(x^n\) in the expansion of the binomial. \((3 x-1)^9, n=2\)

Short answer

The coefficient of \(x^2\) in the expansion \((3x - 1)^9\) is \(111602\).

Step-by-step solution

Determine the Binomial Theorem application parameters

In order to apply the Binomial Theorem to the given binomial \((3x - 1)^9\) to determine the coefficient of \(x^2\), we first need to establish a few values. Here, the binomial is \(a = 3x, b = -1\), and \(n = 9\). Since we're looking for the coefficient of \(x^2\), we want \(k = 2\). This is because the power of \(x\) in the term we're after is determined by the value of \(k\). So we have everything we need to apply the Binomial Theorem.

Apply the Binomial Theorem

Substituting the values established in Step 1 into the Binomial Theorem, we compute \(9C2 * (3x)^{9-2} * (-1)^2 = 36 * (3x)^7\). The coefficient of \(x^2\) is the constant term in this expression.

Calculate the Coefficient

To find the coefficient of \(x^2\), we need to multiply \((3)^7\) by the binomial coefficient \(36\). Doing so gives us \(3^7 * 36 = 111602\).

Key concepts

Binomial Coefficient

When we talk about the binomial coefficient, we're referring to a special number that appears in the binomial theorem. This number helps us find the coefficient of a specific term in a polynomial expansion. Notably, the binomial coefficient is denoted as \( \binom{n}{k} \), and it tells us how many ways we can choose \( k \) items from a set of \( n \) items. This forms the combinatorial heart of the binomial theorem.

In practical terms, the binomial coefficient for our exercise is used to determine the specific term we're interested in within the broader polynomial expansion. For the binomial \((3x - 1)^9\), we used \( \binom{9}{2} \) which equals 36.

To compute a binomial coefficient \( \binom{9}{2} \), the formula is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
• \( \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \)

Use this formula whenever you need to find binomial coefficients—it’s a symmetric and straightforward way to analyze how terms are composed in a polynomial.

Polynomial Expansion

Polynomial expansion involves expressing a binomial raised to a power as a sum of terms. Each term is a product of binomial coefficients, powers of the first term, and powers of the second term. In its essence, using the binomial theorem, we can write any binomial as an expanded polynomial.

Let’s delve into the mechanics of a polynomial expansion using the binomial theorem for the expression \((3x - 1)^9\). When expanded, it takes the form of:

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

For our exercise, \(a = 3x\), \(b = -1\), and \(n = 9\). Of special interest is the power of \(x\). We're looking for the term where \(x\) is raised to the power of 2. The term needed is expressed as:

• \( \binom{9}{2} (3x)^{7} (-1)^2 \)

Following the binomial formula allows us to systematically expand the expression to extract and calculate desired coefficients.

Algebra 2 Concepts

Algebra 2 encompasses many mathematical principles, among them the binomial theorem for polynomial expansions. This theorem is a critical concept within Algebra 2 and is used to address many polynomial-related problems. Understanding its parameters is key to successfully resolving exercises like the one we are examining.

When applying the binomial theorem in Algebra 2, remember these concepts:

• Binomial expressions: Expressions like \((3x - 1)^9\) which involve only two terms
• Exponents: Understanding and manipulating the powers of terms, such as knowing that you're searching for \(x^2\)
• Combinatorics: Utilizing binomial coefficients to count possibilities

These components demonstrate how algebra interplays with real-world problems, translating abstract mathematical rules into clear outcomes. Mastery of Algebra 2 is all about understanding how these parts connect to form comprehensive solutions.