Find the coefficient of the term containing a^4 in the expansion of (3 a-5 x)^12.

Name: Problem 44, Chapter 14: Precalculus
Uploaded: 2020-07-13T20:01:42-08:00
Duration: 3 min 57 s
Channel: Grant Mansfield
Description: Find the coefficient of the term containing a^4 in the expansion of (3 a-5 x)^12.

step 1 Alright, so for this question we are asked to find the coefficient of the term containing A to t

Find the coefficient of the term containing a^4 in the...

Key Concepts

Binomial Theorem

The Binomial Theorem is a formula that allows for the expansion of expressions raised to a power, typically written in the form (a + b)^n. It provides a systematic way of expressing such expansions as a sum of terms involving binomial coefficients and powers of the individual terms. This theorem is central to problems involving the expansion of binomials and the calculation of specific term coefficients.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or (n choose k), represent the number of ways to choose k elements from a set of n elements without regard to the order. In the context of the binomial theorem, these coefficients multiply the respective terms in the expansion. Understanding how to compute and apply these coefficients is crucial for retrieving specific terms in an expanded binomial expression.

General Term in Binomial Expansion

The general term in a binomial expansion is expressed as C(n, k) * a^(n-k) * b^k (or a variant thereof depending on the specific format). This concept is key when extracting a term with a specific power of a variable, as it allows for the identification of the correct term by setting the exponent to match the desired power. Matching the exponent in the general term to the target variable exponent is essential for finding the specific coefficient requested.

Exponent Matching

Exponent Matching involves determining which term in the binomial expansion will contain the desired power of a variable. By comparing the exponent in the general term to the format of the target term (e.g., a^4), one can set up an equation to find the appropriate value of k in the expansion. This process is used to pinpoint the term containing the required variable exponent and subsequently identify its coefficient.

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