Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(4 z^{-1}-3 z\right)^{15} ; \quad \text { last three terms } $$

Step-by-Step Solution

Answer

The last three terms are \(-14348907z^{15}\), \(1074954240z^{13}\), and \(-1965477120z^{11}\).

Step 1: Identify the general term in the expansion

The general term in the binomial expansion of \((a + b)^n\) is given by \(T_k = \binom{n}{k} a^{n-k} b^k\). For the expression \((4z^{-1} - 3z)^{15}\), we have \(a = 4z^{-1}\), \(b = -3z\), and \(n = 15\). Thus, the general term is: \[ T_k = \binom{15}{k} (4z^{-1})^{15-k} (-3z)^k \] Simplifying this gives: \[ T_k = \binom{15}{k} 4^{15-k} (-3)^k z^{-15+k} z^k = \binom{15}{k} 4^{15-k} (-3)^k z^{-15+2k} \]

Step 2: Identify the exponents needed for the last three terms

The exponent in the term is \(-15 + 2k\). For the last three terms of the expansion, we choose values of \(k\) such that \(-15 + 2k\) results in the smallest possible non-negative exponents close to zero. This implies using the largest possible values for \(k\), i.e., \(k = 15\), \(k = 14\), and \(k = 13\).

Step 3: Calculate the terms for k = 15, 14, and 13

1. \(k = 15\): \[ T_{15} = \binom{15}{15} 4^{0} (-3)^{15} z^{-15 + 30} = (-3)^{15} z^{15} = -14348907z^{15} \]2. \(k = 14\): \[ T_{14} = \binom{15}{14} 4^{1} (-3)^{14} z^{-15 + 28} = 15 \cdot 4 \cdot 3^{14} z^{13} = 1074954240z^{13} \]3. \(k = 13\): \[ T_{13} = \binom{15}{13} 4^{2} (-3)^{13} z^{-15 + 26} = 105 \cdot 16 \cdot (-3)^{13} z^{11} \approx -1965477120z^{11} \]

Step 4: Formula simplification

Now we simplify the terms to get the coefficients of the last three terms:1. For \(k = 15\), the term is \(-14348907z^{15}\).2. For \(k = 14\), the term is \(1074954240z^{13}\).3. For \(k = 13\), the term is approximately \(-1965477120z^{11}\) if simplified further.

Key Concepts

Understanding the Binomial Theorem

The binomial theorem is an essential mathematical concept used to expand expressions raised to a power. It provides a formula to express

• the powers of sums \((a + b)^n\)
• as the sum of terms of the form \(\binom{n}{k} a^{n-k} b^k\)

where \(\binom{n}{k}\) are the binomial coefficients. This theorem allows us to calculate individual terms without having to expand the entire expression. In our example, the expression \((4z^{-1} - 3z)^{15}\), is expanded using the binomial theorem to find just the last three terms. By knowing the components \(a\), \(b\), and \(n\), you can directly calculate the desired terms.

Exponent Calculation in Expressions

When dealing with powers and exponents in expressions like \((4z^{-1} - 3z)^{15}\), calculating the exponents is crucial. For each term in the expansion, the exponents are determined by the specific value of \(k\), the term number. The rule for any term's exponent calculation in the binomial expansion is based on

• the formula \(-15 + 2k\)

This formula helps to adjust each term's power by identifying the right \(k\) needed to achieve the smallest possible non-negative exponents. In this exercise, the focus was on choosing \(k=15, 14, 13\) to simplify to the smallest exponents for the last three terms.

Combinatorial Coefficients Made Simple

Combinatorial coefficients, often called binomial coefficients, are a central element of the binomial expansion formula. These coefficients are denoted by \(\binom{n}{k}\) and are calculated using the formula \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]For example, for \(n=15\) and different values of \(k\) such as \(15, 14,\) and \(13\), the coefficients \(\binom{15}{15}\), \(\binom{15}{14}\), and \(\binom{15}{13}\) serve to scale the different terms in the expansion. These coefficients represent the number of ways to choose \(k\) elements from a set of \(n\) elements, ensuring that each expansion term has the correct multiplier.

Simplifying Algebraic Expressions

In algebra, simplifying expressions involves reducing complex terms to simpler forms to make calculations more manageable. Following the expansion using the binomial theorem, simplifying the terms is essential. Here,

• each term, \(\binom{15}{k} 4^{15-k} (-3)^k z^{-15+2k}\), is computed separately
• then simplified to yield the last three terms of the expansion: \(-14348907z^{15}\), \(1074954240z^{13}\), and \(-1965477120z^{11}\)

Simplification involves calculating powers and substitutions, ensuring each term is presented in its simplest form for ease of understanding and application.