Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. \(\left(3 x^{2}-y^{3}\right)^{10}\) fourth term

Step-by-Step Solution

Answer

The fourth term is \(-262440 x^{14} y^{9}\).

Step 1: Recognize the Binomial Theorem

The expansion of an expression in the form \((a + b)^n\) is given by the binomial theorem: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Where \(\binom{n}{k}\) is the binomial coefficient. In this problem, \(a = 3x^2\), \(b = -y^3\), and \(n = 10\).

Step 2: Identify the Formula for the Fourth Term

The fourth term in the expansion of a binomial expression can be found using the general formula for the \((k+1)\)-th term: \[T_{k+1} = \binom{n}{k} a^{n-k} b^k\]For the fourth term, set \(k = 3\) because the \(k+1 = 4\).

Step 3: Calculate the Binomial Coefficient

Calculate the binomial coefficient for the fourth term where \(n = 10\) and \(k = 3\): \[\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120\]

Step 4: Substitute Values into the Formula

Use the values of \(a = 3x^2\) and \(b = -y^3\) for the fourth term: \[T_4 = \binom{10}{3} (3x^2)^{10-3} (-y^3)^3\]This simplifies to: \[T_4 = 120 (3x^2)^7 (-y^3)^3\]

Step 5: Simplify the Fourth Term Expression

Simplify \((3x^2)^7\) and \((-y^3)^3\):\[(3x^2)^7 = 3^7(x^2)^7 = 2187x^{14}\]\[(-y^3)^3 = -y^9\]Thus, the fourth term expression is: \[T_4 = 120 \times 2187x^{14} \times (-y^9)\]

Step 6: Final Calculation of the Fourth Term

Calculate the final value for the fourth term by combining constants and variable exponents:\[ T_4 = 120 \times 2187 \times (-1)x^{14}y^{9}\]\[T_4 = -262440 x^{14} y^{9}\]

Key Concepts

Binomial Expansion

When we talk about binomial expansion, we are dealing with expressions raised to a power, like \((a + b)^n\). The binomial theorem provides a way to expand these expressions without multiplying them out directly.
Imagine having something like \((3x^2 - y^3)^{10}\), which can be a handful to expand completely. The binomial theorem helps simplify this by telling us how each term in the expansion is structured.
When you expand a binomial, each term in the expansion takes the form \(T_k = \binom{n}{k} a^{n-k} b^k\). This means we use binomial coefficients and powers of \(a\) and \(b\) to construct each term. This method ensures you can focus on specific parts of the expression, like the fourth term, without having to find the entire expansion.

Binomial Coefficient

Binomial coefficients are crucial to the binomial theorem. They appear as \(\binom{n}{k}\) and determine the number of ways \(k\) items can be chosen from \(n\) items. In our example, you compute it for \(n = 10\) and \(k = 3\), which means we're interested in the fourth term, so it becomes:
\[\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120\]
This value tells us how many times each part of our expanded expression will appear. It's a fascinating blend of algebra and combinatorics. The coefficient also impacts the magnitude of each term in the expansion. So for \(\binom{10}{3} = 120\), it significantly scales our result, indicating that this term will stack up higher in value.

Polynomial Expansion

Polynomial expansion is a method used to express a binomial raised to a power in the form of a polynomial. Each term in a polynomial has coefficients and variables raised to some powers.
In our exercise, we're handling the binomial \((3x^2 - y^3)^{10}\), and expanding it forms a polynomial comprised of terms like the fourth one we've found: \(-262440x^{14}y^{9}\).
The polynomial expansion allows us to understand how each variable contributes to the expression. Through polynomial expansion, the powers in the terms are systematically arranged, showing us how both \(x\) and \(y\) influence each other within each term. The process bridges simple binomials with complex expressions, making them manageable.

Exponential Expressions

Exponential expressions are expressions where numbers or variables are raised to powers as exponents, like \((3x^2)^{10}\). They are central to both binomial and polynomial expansions.
In our case, the expression \((3x^2)^{7}\) becomes \(2187x^{14}\), showing how the exponent is distributed over products. With exponential expressions, it's simpler to follow how powers combine or break down, especially in complex terms.
Understanding these exponents helps in simplifying expressions, as seen when handling terms like \((-y^3)^{3}\), which transforms into \(-y^9\). Exponential handling becomes a strategy for dealing with large powers efficiently while maintaining arithmetic logic throughout the expansion process.