Problem 44 - College Algebra

Question

Find the term indicated in each expansion. \(\left(x^{3}+y^{2}\right)^{8} ;\) sixth term

Step-by-Step Solution

Answer

The sixth term in the expansion of \(\left(x^3+y^2\right)^8\) is \(56 * x^9 * y^{10}\).

Step 1: Identify the variables and the power in the binomial theorem

From the binomial \(\left(x^3+y^2\right)^8\), identify that \(a=x^3\), \(b=y^2\) and \(n=8\). The binomial formula states that any term in the expansion is given by \((a+b)^n= \binom{n}{k} a^{n-k} b^k\). We are tasked with finding the sixth term.

Step 2: Calculate the binomial coefficient for the sixth term

The formula for the binomial coefficient gives the binomial coefficient of 'r+1'th term in the expansion of \((a+b)^n\) as \(\binom{n}{r}\). This indicates that the 'r' in our case will be '6-1' or 5. Using this, we can find the binomial coefficient for the sixth term as \(\binom{8}{5}=56\).

Step 3: Apply the binomial theorem

Now that we have the binomial coefficient, we can find the sixth term by applying the binomial theorem. The sixth term will be \(\binom{n}{k} a^{n-k} b^k= \binom{8}{5} (x^3)^{8-5} (y^2)^5 = 56 * (x^3)^3 * (y^2)^5\). Finally, simplify to get the sixth term.

Step 4: Simplify the sixth term

Before the final calculation, let's simplify everything. This should be a simple substitution where we multiply the binomial coefficient with the powers of \(x\) and \(y\). After simplifying, the sixth term is \(56 * x^9 * y^{10}\).

Key Concepts

Binomial Coefficient

In the context of the binomial theorem, a binomial coefficient is a crucial component that helps determine the specific terms in a binomial expansion. Given a binomial expression of the form
\((a + b)^n\), you can find any term in the expansion using the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

These coefficients are systematically arranged in what is known as Pascal's Triangle. Each coefficient gives us the number of ways to select \(k\) items from \(n\) without regard to the order of selection.
In the exercise, we found the coefficient for the sixth term by determining \(\binom{8}{5}\), which equals 56. This number helps us calculate the contributions of each term we want to find in the expansion. Calculating these coefficients accurately is key to understanding the overall polynomial expansion in binomial expressions.

Polynomial Expansion

Polynomial expansions refer to expressing a power of a binomial expression as a sum of terms. Each term in the expansion results from a recursive application of the distributive property, specifically the binomial theorem:

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

This formula allows us to expand and find specific terms in powers of binomials, such as \((x^3 + y^2)^8\). It simplifies complex expressions into manageable terms.
In our example, the sixth term is derived from raising the terms of the binomial to various powers determined by the binomial coefficient calculated previously, leading to the expression \(56 \times (x^3)^3 \times (y^2)^5\). This systematic use of the formula gives us a way to handle large powers common in algebraic manipulation.

Algebraic Expressions

Algebraic expressions play a foundational role in mathematics by representing quantities in terms of variables and constants. In the binomial theorem, algebraic expressions are raised to powers and expanded. This requires simplifying
using rules of exponents and coefficients.

• An algebraic expression follows standard operations: addition, subtraction, multiplication, and division.
• When working with algebraic expressions like \((x^3+y^2)^8\), understanding variables, constants, and their combinations is crucial.

The final task in our problem is simplifying the expression \(56 \times x^9 \times y^{10}\), taking each component from the expansion and combining them. By understanding the nature of algebraic expressions, students can effectively expand and simplify even complex binomial expressions.