Problem 45 - Algebra 2

Question

Suppose \(_{8} \mathrm{C}_{3} x^{5} y^{3}\) is a term of a binomial expansion. Write the next term.

Step-by-Step Solution

Answer

The next term is \( _{8}C_{4} x^{4} y^{4} \)

Step 1: Understand the Binomial Theorem Components

In the binomial expansion of \( (x + y)^n \) the general term is given by \( _{n}C_{r} x^{n-r} y^{r} \). Here, the given term \( _{8}C_{3} x^{5} y^{3} \) represents the fourth term (since \( r \) starts from 0) of the expansion \( (x + y)^8 \) where \( r = 3 \) and \( n = 8 \) implying that \( n-r = 5 \) and the powers of \( x \) and \( y \) are 5 and 3 respectively.

Step 2: Identify the Next Term's 'r' Value

To find the next term in the sequence, we increment the value of \( r \) by 1. Since the given term has \( r = 3 \) for \( _{8}C_{3} \), the next term will have \( r = 4 \) in \( _{8}C_{r} \) which reflects how many positions we've moved from the beginning of the expansion.

Step 3: Determine the Next Term

The next term will have \( r \) increased by 1 to become 4, and therefore \( n-r \) will decrease by 1 to become 4. Therefore, the next term is given by \( _{8}C_{4} x^{8-4} y^{4} = _{8}C_{4} x^{4} y^{4} \) since the exponents of \( x \) and \( y \) in consecutive terms of a binomial expansion always sum to \( n \) (in this case, 8).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that describes the expansion of powers of a binomial expression. A binomial is a polynomial with two terms, typically written as \( (a + b)^n \), where \( n \) is a non-negative integer. The theorem provides a formula to expand expressions of this form into a sum involving terms of the form \( a^{n-k}b^k \).

• The expansion follows a specific pattern where each term is a combination of the two variables raised to complementary powers that always add up to \( n \).
• It is particularly important when dealing with higher powers of a binomial, as manually expanding them would be tedious and prone to error.
• The theorem also introduces the concept of binomial coefficients, which are integral in understanding not just binomial expansion, but also combinations in probability and other areas of mathematics.

By using the Binomial Theorem, complex polynomial expansions become much more manageable, and patterns within the coefficients can be explored. For the term \( _{8}\mathrm{C}_{3}x^{5}y^{3} \), according to the theorem, it is part of the expansion of \( (x + y)^8 \) and is the fourth term in the sequence.

Combination Notation

Combination notation, often represented as \( _n\mathrm{C}_r \), \( C(n, r) \) or \( \binom{n}{r} \), is a way of expressing the number of ways to choose \( r \) elements from a bigger set of \( n \) elements without taking the order into account. It's an essential concept in combinatorics, a branch of mathematics that deals with counting, arrangement, and combination of elements.

• Mathematically, combination notation is defined using factorial notation as \( _n\mathrm{C}_r = \frac{n!}{r!(n - r)!} \) where \( n! \) (read as 'n factorial') is the product of all positive integers up to \( n \).
• Combinations are central to the Binomial Theorem because each term in a binomial expansion is preceded by a binomial coefficient, which is a specific combination \( _n\mathrm{C}_r \) that corresponds to that term.

In our given term \( _{8}\mathrm{C}_{3} \), this signifies the number of ways to choose 3 elements from a set of 8, which is also the coefficient that precedes \( x^{5}y^{3} \) in the expansion of \( (x + y)^8 \) as indicated by the theorem.

Polynomial Terms

In algebra, a polynomial is an expression consisting of variables (also known as indeterminates), coefficients, and the operations of addition, subtraction, multiplication, non-negative integer exponents of variables. Polynomial terms are particular components of a polynomial that are separated by the plus or minus signs.

• Terms of a polynomial can be of various degrees, which are determined by the sum of the exponents of the variables within them.
• For example, in the polynomial \( 5x^3 - 2xy^2 + 7 \), \( 5x^3 \) is a term of degree 3, \( -2xy^2 \) is a term of degree 3 (\( 1+2 \) for \( x \) and \( y \) respectively), and \( 7 \) is a term of degree 0.

In the context of binomial expansion, each term of the expanded form will be a polynomial term consisting of a binomial coefficient and the two variables from the original binomial raised to specific powers.

Binomial Coefficients

Binomial coefficients are the numerical factors that multiply the different terms in the expansion of a binomial expression like \( (x + y)^n \). They play a crucial role as they determine the relative weight of each term in the expansion.

• These coefficients are symmetric, meaning the \( k \)th coefficient in the expansion is equal to the \( n - k \)th coefficient, reflecting the property \( _n\mathrm{C}_k = _n\mathrm{C}_{n - k} \).
• The coefficients form a well-known arrangement called Pascal's Triangle, where each number is the sum of the two directly above it.
• The binomial coefficient for any term in a binomial expansion can be determined by the combination formula and is represented as \( _n\mathrm{C}_r \) or \( \binom{n}{r} \) where \( r \) is the term number in the expansion.

In the example given \( _{8}\mathrm{C}_{3}x^{5}y^{3} \) represents a binomial coefficient for the fourth term of \( (x + y)^8 \) since counting starts from zero, and illustrates the importance of these coefficients in identifying the terms of a binomial expansion.