Problem 43 - A Graphical Approach to College Algebra

Question

Write the indicated term of each binomial expansion. Fifteenth term of \(\left(x-y^{3}\right)^{20}\).

Step-by-Step Solution

Answer

The fifteenth term is \(38760 x^6 y^{42}\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For our problem, \(a = x\), \(b = -y^3\), and \(n = 20\).

Step 2: Identify the General Term

The general term in the expansion is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). Here, we substitute \(a = x\), \(b = -y^3\), and \(n = 20\):\[T_{k+1} = \binom{20}{k} x^{20-k} (-y^3)^k\]

Step 3: Find the Fifteenth Term

Since we want the 15th term, set \(k + 1 = 15\). Thus, \(k = 14\). Substitute \(k = 14\) into the general term formula:\[T_{15} = \binom{20}{14} x^{20-14} (-y^3)^{14}\]

Step 4: Simplify the Expression

Calculate each part of the expression:- Compute \(\binom{20}{14}\).- Simplify \(x^{20-14} = x^6\).- Simplify \((-y^3)^{14} = (-1)^{14} (y^3)^{14} = y^{42}\) (since \((-1)^{14} = 1\)).So, \[T_{15} = \binom{20}{14} x^6 y^{42}\]

Step 5: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{20}{14}\) can be calculated as:\[\binom{20}{14} = \frac{20!}{14!6!} = 38760\]

Step 6: Write the Final Term

Substitute the binomial coefficient into the expression from Step 4:\[T_{15} = 38760 x^6 y^{42}\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra that describes the expansion of powers of a binomial expression. A binomial is simply an algebraic expression containing two terms, such as \((a + b)\). According to the Binomial Theorem, any power \((a + b)^n\) can be expanded into a sum involving terms of the form \(a^{n-k}b^k\), where \(k\) is a non-negative integer ranging from 0 to \(n\). The coefficient of each term is given by the binomial coefficient \(\binom{n}{k}\), which counts the number of ways to choose \(k\) items from a set of \(n\) items. This explains why the theorem has connections to combinatorics.

The formula for the binomial theorem is:

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

This expands any binomial raised to a power \(n\) into \(n+1\) terms, starting from the 0th power up to the nth power. Each expansion term involves multiplying binomial coefficients with the appropriate powers of \(a\) and \(b\).

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. When handling binomial expansions, combinatorics plays a critical role during the calculation of binomial coefficients. The binomial coefficient \(\binom{n}{k}\) represents the number of combinations where \(k\) items can be chosen from \(n\) items without caring about the order. It is calculated using the formula:

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

In the context of the binomial expansion problem, combinatorics helps in computing these coefficients efficiently. For example, to find the coefficient for any specific term in the expansion of \((x-y^3)^{20}\), like the fifteenth term, combinatorial computations are necessary.

These coefficients enable us to understand how often specific products of the binomial terms appear in the expanded form. Using factorials, combinatorics gives a clear method for determining these coefficients, which can then be plugged back into the general term of the Binomial Theorem.

Polynomial Algebra

Polynomial algebra deals with expressions known as polynomials, which are sums of terms consisting of variables and coefficients. Each polynomial's terms have a variable raised to a power. The binomial theorem helps expand binomials into polynomials. For example, the binomial expression \((x - y^3)^{20}\) becomes a polynomial with multiple terms when expanded.

In polynomial algebra, understanding the structure of polynomials is important. A polynomial's degree is the highest power of its variable, and each term in a polynomial is made of a coefficient and one or more variables raised to specific powers. The challenge often lies in identifying specific terms, like when specifying that we're interested in the fifteenth term of the binomial expansion. Using the Binomial Theorem, we derive and simplify terms like \(38760 x^6 y^{42}\), ensuring all parts of each term, both variable and coefficient, are correctly calculated and placed within the polynomial.

By using polynomial algebra, not only can we expand binomials, but we can also simplify, add, subtract, and multiply complex polynomial expressions.

Exponential Expressions

Exponential expressions are types of mathematical expressions that involve exponents, where bases are raised to the power of exponents. In the context of binomial expansions, each term involves exponential expressions both for the binomial variables and for their respective powers.

In the problem of expanding \((x - y^3)^{20}\), exponential expressions emerge throughout the process. Each term of the expansion formula \(T_{k+1}\) involves powers of the binomial's components. For instance, in the fifteenth term \(T_{15} = 38760 x^6 y^{42}\), the exponents 6 and 42 indicate the polynomial's degree for each variable respectively. Here, \(x^{20-k}\) and \((-y^3)^k\) highlight the exponential nature of binomial terms after being expanded.

Understanding and manipulating these expressions involves simplifying exponential powers, where laws such as \(a^m \cdot a^n = a^{m+n}\) may be applied. Recognizing how exponents distribute across terms helps implement operations efficiently, making calculations in polynomial expansions more straightforward. Expressing complex powers neatly and understanding their algebraic properties form the foundation of dealing with exponential expressions in algebraic contexts.