Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The third term of \((6 x-3 y)^{7}\)

Step-by-Step Solution

Answer

The third term is \(1469664x^5y^2\).

Step 1: Identify the required formula

To find a specific term from a binomial expansion such as \((a + b)^n\), we use the formula for the \((r+1)^{th}\) term: \( T_{r+1} = \binom{n}{r} a^{n-r} b^r \). Here, \(n = 7\) and we want the third term, so \(r = 2\).

Step 2: Substitute values into the formula

Let's determine the third term using \( T_{3} = \binom{7}{2} (6x)^{7-2} (-3y)^2 \). We have \(a = 6x\), \(b = -3y\), \(n = 7\), and \(r = 2\).

Step 3: Calculate the binomial coefficient

Calculate \(\binom{7}{2} = \frac{7!}{2!(7-2)!} = 21\).

Step 4: Compute the powers of a and b

Compute \((6x)^{5} = 7776x^5\) and \((-3y)^{2} = 9y^2\).

Step 5: Multiply results to find the term

Multiply the binomial coefficient by the powers: \(21 \times 7776x^5 \times 9y^2 = 1469664x^5y^2\).

Key Concepts

Binomial Expansion

Binomial Expansion is a fundamental concept in algebra that allows us to expand expressions raised to a power. When you see an expression like \((a + b)^n\), it can be expanded into a series of terms. Each term in this expansion is a result of multiplying the base elements\(a\) and \(b\) raised to specific powers. This is made possible by following the pattern established by the Binomial Theorem. This theorem not only provides a way of expanding binomials but also gives us a structure we can use mathematically to solve various problems involving polynomial expressions.Expanding a binomial expression manually could be challenging especially for larger exponents. However, understanding the relationship between the coefficients and terms helps simplify the process. By breaking down the expansion into the product of coefficients and variables raised to their respective powers, this often intricate task becomes manageable and applicable to problem-solving situations.

Binomial Coefficient

The Binomial Coefficient plays a key role in determining the specific terms of a binomial expansion. Represented by \(\binom{n}{r}\), it calculates the number of ways to choose \(r\) elements from \(n\) elements. You'll often see it in the formula for the \((r+1)^{th}\) term in a binomial expansion, which is:

• \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\)

In our example, we calculated the binomial coefficient \(\binom{7}{2}\) using the formula:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]It resulted in 21, showing how many unique ways the terms from the binomial can combine to form each specific term of the sequence. This process underscores the combinatorial nature of binomial expansions.

Power of a Binomial

The Power of a Binomial refers to raising a binomial to a particular exponent. Each term in the binomial expansion is derived by calculating specific powers of its components, which gives meaning to the expression \((a + b)^n\).Let's consider the binomial \((6x - 3y)^7\). The power of each term in the expansion is determined by the exponents applied to \(a\) and \(b\). For the third term in our example, the expression \( (6x)^{5} \) translates to utilizing powers:

• First component: \((6x)\) raised to \(5\)
• Second component: \((-3y)\) raised to \(2\)

The resulting third term from combining these computations via multiplication is simplified to \(1469664x^5y^2\).Understanding how to compute and apply these powers is essential. It makes managing expansions and extracting specific terms much simpler, particularly as problems become more complex or require more terms.