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Chapter 11: Problem 352

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The fourth term of \((3 x-2 y)^{5}\)

Short Answer

Expert verified
The fourth term is \(-720x^2y^3\).

Step by step solution

01

Identify the Term Formula

To find a specific term in the expansion of a binomial expression \((a + b)^n\), use the formula for the \((k+1)\)-th term, which is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). We want to find the fourth term, so \(k = 3\).
02

Apply the Known Values

Apply the known values to the formula. In this problem, \(a = 3x\), \(b = -2y\), and \(n = 5\). Let \(k = 3\) to find the fourth term: \(T_4 = \binom{5}{3} (3x)^{5-3} (-2y)^3\).
03

Calculate the Binomial Coefficient

Calculate the binomial coefficient \(\binom{5}{3}\), which is \( \frac{5 \times 4}{2 \times 1} = 10\).
04

Compute the Powers of the Binomial Terms

Compute the powers: \((3x)^{5-3} = (3x)^2 = 9x^2\) and \((-2y)^3 = -8y^3\).
05

Multiply the Components

Combine all components to find the coefficient for the fourth term: \(T_4 = 10 \times 9x^2 \times (-8y^3) = -720x^2y^3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Coefficients
Binomial coefficients play a crucial role when expanding expressions raised to a power, notably in binomial expansions. These coefficients are numerical factors that determine the weight of each term within the expansion. The binomial coefficient is denoted as \( \binom{n}{k} \), which is read as "n choose k". It is calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n! \) represents the factorial of n. The factorial function \( n! \) is the product of all positive integers up to \( n \).
  • For example, to find \( \binom{5}{3} \), you calculate it as \( \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \).
  • This coefficient indicates how many combinations of 3 items can be selected from a set of 5 items.
Understanding these coefficients helps in determining the distribution of polynomial terms without writing out entire expansions.
Polynomial Expansion
The polynomial expansion process involves breaking down expressions of the form \((a + b)^n\) into a sum of terms, each multiplied by a binomial coefficient. The Binomial Theorem provides a formula to quickly calculate these expansions. Each term in the expansion can be written as \( \binom{n}{k} a^{n-k} b^k \). This formula takes into account:
  • The binomial coefficient, \( \binom{n}{k} \), which decides the numeric multiplier.
  • The decreasing powers of \( a \) and increasing powers of \( b \) as \( k \) increases from 0 to \( n \).
By applying the theorem, you can efficiently expand high powers of binomials. This is particularly useful in finding specific terms without dealing with the entire polynomial. So, if you are asked to find a particular term such as the fourth term in \((3x - 2y)^5\), applying the polynomial expansion formula saves a lot of time and effort.
Term Calculation
Term calculation within a binomial expansion focuses on finding terms quickly using the Binomial Theorem. Instead of expanding the entire expression, you can directly calculate any term using the formula \( T_{k+1} = \binom{n}{k} a^{n-k} b^k \).
Here's a step-by-step way to compute a specific term, such as the fourth term in \((3x - 2y)^5\):
  • Set the variables: Identify \( a = 3x \), \( b = -2y \), and \( n = 5 \). To find the fourth term, use \( k = 3 \) since the term formula is \( T_{k+1} \).
  • Compute the binomial coefficient: \( \binom{5}{3} = 10 \).
  • Calculate the powers: Evaluate \((3x)^{5-3} = (3x)^2 = 9x^2 \) and \((-2y)^3 = -8y^3 \).
  • Final calculation: Multiply all components: \( T_4 = 10 \times 9x^2 \times (-8y^3) = -720x^2y^3 \).
This method is both efficient and precise for finding any desired term in a binomial expansion.

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