Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(3 y^{3}-2 x^{2}\right)^{4} ; \quad \text { term that contains } y^{9} $$

Step-by-Step Solution

Answer

The term containing \( y^9 \) is \( -216x^2y^9 \).

Step 1: Understanding the Problem

We are asked to find the term in the expansion of \( (3y^3 - 2x^2)^4 \) that contains \( y^9 \). This involves identifying specific terms in a binomial expansion.

Step 2: Apply Binomial Theorem

The Binomial Theorem states \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \( a = 3y^3 \), \( b = -2x^2 \), and \( n = 4 \). We are looking for a term that includes \( y^9 \).

Step 3: Find the Power of \( y^3 \)

In the term \( (3y^3)^{n-k} \), the exponent of \( y \) will be \( 3(n-k) \). For the term to contain \( y^9 \), set \( 3(n-k) = 9 \). Solving gives \( n-k = 3 \). Thus, \( k = 1 \) since \( n = 4 \).

Step 4: Calculate the Term with \( k = 1 \)

Substitute \( k = 1 \) into the binomial term formula. So, the term is \( \binom{4}{1} (3y^3)^{3} (-2x^2)^{1} \).

Step 5: Simplify the Term

Calculate each part: \( \binom{4}{1} = 4 \), \( (3y^3)^3 = 27y^9 \), and \( (-2x^2)^1 = -2x^2 \). Combine to form the term: \( 4 \cdot 27y^9 \cdot (-2x^2) = -216x^2y^9 \).

Key Concepts

Term Identification

Identifying specific terms in a binomial expansion is a crucial step when solving problems with expressions like \((3y^3 - 2x^2)^4\). Here, the goal is to pinpoint the term that contains a particular component, such as \(y^9\).

• In a binomial expression, two distinct terms raised to a power, such as \(a\) and \(-b\), are expanded to form various terms.
• Each term is a combination of the powers of \(a\) and \(b\), multiplied by a specific binomial coefficient.

To find a specific term containing \(y^9\), focus on the terms where \(y\) appears. By setting the power of \(y^3\) equal to 9, we identify the term's position without fully expanding the expression.
This approach allows us to efficiently narrow down the search to terms meeting the desired criteria.

Polynomial Expansion

Polynomial expansion involves taking a simple expression with two terms and expressing it as a sum of multiple terms. Using the Binomial Theorem, we expand expressions of the form \((a + b)^n\).

• The Binomial Theorem uses the formula: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).
• In our problem, the expression \((3y^3 - 2x^2)^4\) needs to be expanded using this theorem.

Each term in the expanded polynomial is a product of the binomial coefficient, powers of the first term \(a\), and powers of the second term \(-b\).
This method allows us to express complex algebraic expressions as polynomials, where each term corresponds to a specific arrangement of factors and powers.

Exponent Calculation

Calculating exponents is an important part of expanding polynomials using the Binomial Theorem. Exponents tell us how many times a base is multiplied by itself. In the context of the expression \((3y^3 - 2x^2)^4\), each term in the expansion involves calculating exponents for the binomial components.

• The power of each component, such as \((3y^3)\) or \((-2x^2)\), is determined by the values of \(n-k\) and \(k\) respectively.
• Solved by setting \(3(n-k) = 9\) to find the desired \(y^9\) term, we calculate: \((3y^3)^{3} = 27y^9\).

This involves understanding that exponents must be correctly calculated to find the correct power contributions of each variable.
Exponent calculation ensures that each term in the polynomial expression reflects the accurate power of each variable.