Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The fourth term of \((2 x-3 y)^{4}\)

Step-by-Step Solution

Answer

The fourth term is \(-216xy^3\).

Step 1: Understand the Problem

We need to find the fourth term of the binomial expansion \((2x - 3y)^4\) without fully expanding it. This problem involves using the binomial theorem.

Step 2: Know the Binomial Theorem

The binomial theorem states that for any positive integer \(n\), \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\).In this case, \(a = 2x\), \(b = -3y\), and \(n = 4\).

Step 3: Identify the Fourth Term

The \(k\)-th term in the binomial expansion is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\).For the fourth term, \(k = 3\).

Step 4: Calculate the Binomial Coefficient

Find the binomial coefficient \(\binom{4}{3}\):\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4. \]

Step 5: Calculate the Fourth Term

Use the values: \(a = 2x\), \(b = -3y\), \(k = 3\), and \(n = 4\):- The coefficient is \(\binom{4}{3} = 4\).- Substitute into the formula:\[ T_4 = \binom{4}{3} (2x)^{4-3} (-3y)^3 = 4 \times (2x)^{1} \times (-3y)^3. \]- Simplify:\[ T_4 = 4 \times 2x \times (-27y^3) = -216xy^3. \]

Key Concepts

Binomial Expansion

The concept of binomial expansion is rooted in a fundamental principle within algebra known as the binomial theorem. This theorem provides a clear pathway to expand expressions of the form \((a + b)^n\) without having to multiply the expression by itself repeatedly. The equation \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\) reveals that the expansion is simply the sum of a series of terms. Each term involves a combination of coefficients and powers of the variables involved.In practical terms, binomial expansion allows us to find specific terms in a polynomial expansion without full multiplication. This is what makes it incredibly useful in various mathematical and practical applications—be it calculus, probability theory, or even computer algorithms. For instance, in the context of the expression \((2x - 3y)^4\), the binomial theorem facilitates quickly pinpointing specific terms, such as the fourth term, without heavily computing the entire equation.

Binomial Coefficient

The binomial coefficient is a key player in the binomial expansion. It appears in the term \(\binom{n}{k}\), which is read as "n choose k" and represents the number of ways to choose k items from n items without regard to order. This coefficient is calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where "!" denotes a factorial, meaning the product of all positive integers up to the given number.Using our example \((2x - 3y)^4\), if we want to find the fourth term, we need a coefficient that corresponds to the situation where three terms have been chosen out of four (because the first term we look for is the zero-th, corresponding to the k+1 notation). Therefore, we compute \(\binom{4}{3}\), which yields:

• \(\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4\)

This value is then used to calculate a specific term in the expansion, showing its critical role in determining the contribution of each term in the binomial series.

Algebra

Algebra serves as the foundational framework through which concepts like the binomial theorem can be understood and applied. It involves working with variables and constants to form equations and expressions, enabling systematic problem-solving and manipulation of mathematical statements.In the context of the exercise \((2x - 3y)^4\), algebra allows us to solve for specific terms without fully evaluating the expression. Through understanding variables as placeholders, such as \(a = 2x\) and \(b = -3y\), and applying algebraic rules, students learn to deftly handle complex equations. These skills in algebra are not simply limited to expanding binomials; they serve across multiple topics in mathematics, showing the versatility and power of algebraic thinking in solving real-world problems and more advanced mathematical challenges.