Problem 39 - A Graphical Approach to College Algebra

Question

Write the indicated term of each binomial expansion. Sixth term of \((4 h-j)^{8}\).

Step-by-Step Solution

Answer

The sixth term in the expansion is \(-3584h^3j^5\).

Step 1: Identify the formula for binomial expansion

The general term in the expansion of a binomial expression \((a + b)^n\) is given by the formula: \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]where \(T_{k+1}\) is the \((k+1)\)-th term, \(\binom{n}{k}\) is the binomial coefficient, n is the exponent, and a and b are the terms of the binomial.

Step 2: Substitute values to find the 6th term

Identify \(a = 4h\), \(b = -j\), and \(n = 8\). To find the 6th term, set \(k = 5\) (since \(k+1 = 6\)). Substitute these into the formula:\[ T_{6} = \binom{8}{5} (4h)^{8-5} (-j)^5 \]

Step 3: Calculate binomial coefficient

Calculate the binomial coefficient for \(\binom{8}{5}\): \[ \binom{8}{5} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

Step 4: Compute powers of terms

Calculate the powers of the terms:\((4h)^{8-5} = (4h)^3 = 4^3 \times h^3 = 64h^3\) \((-j)^5 = (-1)^5 \times j^5 = -j^5\).

Step 5: Multiply and simplify the expression

Substitute these back into the expression for the 6th term:\[ T_6 = 56 \times 64h^3 \times (-j^5) \]Simplify the multiplication:\[ T_6 = 56 \times 64 \times h^3 \times (-j^5) = -3584h^3j^5 \]

Key Concepts

Understanding the Binomial Theorem

The binomial theorem is a powerful mathematical tool used to expand expressions that are raised to a positive integer power. It allows us to express \((a + b)^n\) as a sum of terms involving coefficients, powers of \(a\), and powers of \(b\). This theorem is especially useful in algebra and calculus, providing a systematic method for expansion. Let's delve deeper into this concept:

• Each term in the expansion takes the form \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is called the binomial coefficient.
• The binomial coefficient is calculated using combinatorics, specifically as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), representing ways to choose \(k\) elements from \(n\) elements without regard to order.
• In practice, this theorem allows us to find specific terms or even the entire expanded form of the binomial expression without writing out each individual product and sum.

Using this theorem, complex algebraic expressions become simpler to handle. It becomes particularly interesting as the powers of \(n\) become larger, providing vast and varied applications in mathematical fields.

Exploring Combinatorics

Combinatorics is the branch of mathematics focused on counting, arrangement, and combination. It's an essential component in calculating binomial coefficients. Let's explain this in the context of the binomial theorem:

• A binomial coefficient, \(\binom{n}{k}\), is a central part of combinatorics. It determines how many different ways we can select \(k\) items from a total of \(n\) items.
• These coefficients are used in the binomial expansion to weigh each term correctly, reflecting the number of ways to arrange components in each position.
• The use of factorial notation, \( n! \) (pronounced "n factorial"), simplifies these calculations, where \(n! = n \times (n-1) \times (n-2) \times ... \times 1\).
• Combinatorics helps us in not only mathematical computations but also in practical problem-solving activities such as scheduling, data arrangement, and probability.

By applying combinatorial principles, solving problems in binomial expansions becomes a methodical process, enhancing both mathematical understanding and application.

Algebraic Expressions Made Simple

Algebraic expressions involve numbers, variables, and arithmetic operations. They are fundamental in mathematics, representing quantities and relations. In the context of binomial expansions, understanding them is key:

• An algebraic expression like \((4h-j)\) involves variables \(h\) and \(j\), and the constants that multiply these variables, such as the number \(4\).
• When raising an algebraic expression to a power, as in \( (4h-j)^8 \), the binomial theorem provides a formulaic way to expand this algebraically.
• This expansion results in multiple terms where each reflects a specific combination of the powers and the constants expressed through calculations like multiplying \(4^3\) or determining \((-j)^5\).
• Simplifying such expanded algebraic expressions involves multiplying and combining like terms, ensuring the expression can be used effectively in calculus, geometry, or other applications.

By mastering algebraic expressions within the framework of the binomial theorem, students can unlock new levels of problem-solving abilities and deepen their understanding of complex mathematics.