Problem 34 - Algebra and Trigonometry

Question

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{2}\) in the expansion of \((2 x-3)^{9}\)

Step-by-Step Solution

Answer

-314928

Step 1: Understand the Binomial Theorem

The Binomial Theorem states \[ (a + b)^n = \sum_{k=0}^{n} C(n,k) a^{n-k} b^k \]where \(C(n,k)\) is the binomial coefficient, also written as \(\binom{n}{k}\), and defined as \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \].

Step 2: Identify the variables

In the expression \[ (2x - 3)^9 \], a = 2x, b = -3,n = 9.

Step 3: Write the general term

The general term in the expansion of \[ (2x - 3)^9 \] is given by \[ T_k = \binom{9}{k} (2x)^{9-k} (-3)^k \].

Step 4: Find the specific term for \(x^2\)

To find the coefficient of \(x^2\), set the power of \(x\) equal to 2:\[(9-k) = 2.\]Solving for \(k\), we get \[k = 7.\]

Step 5: Substitute back to find the coefficient

Substitute \(k = 7\) into the general term:\[ T_7 = \binom{9}{7} (2x)^{9-7} (-3)^7 \].This simplifies to:\[ T_7 = \binom{9}{7} (2x)^{2} (-3)^7 \].

Step 6: Calculate the binomial coefficient and constants

Calculate \(\binom{9}{7}\):\[\binom{9}{7} = \binom{9}{2} = \frac{9!}{2!7!} = 36.\]Calculate the other constants:\[(2x)^2 = 4x^2\] and \[(-3)^7 = -2187.\]

Step 7: Combine the results

Combining all parts:\[ T_7 = 36 \cdot 4x^2 \cdot (-2187) \] \[ T_7 = 36 \cdot 4 \cdot (-2187) x^2 \]\[ T_7 = -314928 x^2.\]

Step 8: Extract the coefficient

The coefficient of \(x^2\) in the expansion of \((2x - 3)^9\) is \[ -314928. \]

Key Concepts

binomial coefficients

A binomial coefficient is a value that appears in the expansion of a binomial raised to any given power, according to the Binomial Theorem. Binomial coefficients are denoted as \(\binom{n}{k}\) and represent the number of ways to choose k elements from a set of n elements. Mathematically, it is defined as: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]. Here, \(n!\) is the factorial of n, which means the product of all positive integers up to n. For instance, \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\). These coefficients play a crucial role in polynomial expansions and can be directly linked to combinations in probability and combinatorics.

polynomial expansion

Polynomial expansion involves expressing a polynomial in its expanded form. The Binomial Theorem is a tool used for expanding polynomials of the form \((a + b)^n\). The theorem states: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. This means we sum up terms for k ranging from 0 to n. Each term in the expansion includes a binomial coefficient (\(\binom{n}{k}\)), a power of a, and a power of b. For example, expanding \( (x + y)^2 \) yields: \[ (x + y)^2 = \binom{2}{0} x^2 y^0 + \binom{2}{1} x^1 y^1 + \binom{2}{2} x^0 y^2 = x^2 + 2xy + y^2. \]. By following this systematic approach, any binomial expression can be expanded into a sum of terms.

algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of the Binomial Theorem and binomial expansions, algebra helps us understand and solve expressions involving polynomials and coefficients. Concepts such as variables, constants, and the manipulation of expressions are essential in performing these expansions. For instance, in the given problem, algebraic techniques allow us to identify the terms and coefficients, such as determining the general term in the expansion of \((2x - 3)^9\). Specifically, we can find the coefficient of \(x^2\) by solving algebraic equations that determine appropriate values for k. Mastering algebraic skills ensures we accurately apply the Binomial Theorem and resolve complex polynomial expressions efficiently.