Problem 32 - College Algebra

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 \(-2916 x^5 y^2\).

Step 1: Identify the Expression

We need to expand: \((6 x-3 y)^{7}\).

Step 2: Apply the Appropriate Formula

For a binomial square \((a+b)^2\), we use: \((a+b)^2 = a^2 + 2ab + b^2\).
For \((a-b)^2\): \((a-b)^2 = a^2 - 2ab + b^2\).

Step 3: Substitute and Expand

Substituting the values from our expression and expanding, we obtain the result.

Step 4: State the Result

The third term is \(-2916 x^5 y^2\).

Key Concepts

Binomial Expansion

The Binomial Expansion is a method for expressing an algebraic expression that sums two terms raised to a power. When you encounter an expression like \((a + b)^n\), the binomial expansion lets you unfold this into a sum of terms. These terms involve coefficients, which are specific numbers, along with different powers of \(a\) and \(b\).

• The binomial expansion of \((a + b)^n\) follows a specific pattern defined by the Binomial Theorem.
• This theorem explains how to build each term in the expansion according to their respective powers and coefficients.

Instead of multiplying \((a + b)\) by itself \(n\) times, the Binomial Theorem provides a quicker and cleaner solution. Thus, for any positive integer power, you can find the specific term in the series without dealing with a full expansion.

Binomial Coefficient

The Binomial Coefficient plays a crucial role in the binomial expansion. It is represented by the symbol \(\binom{n}{k}\), and it determines the size of each term in the expansion. This coefficient is essentially a number that tells you how many ways you can choose \(k\) elements from a set of \(n\) elements.

• In a term for \((a+b)^n\), \(\binom{n}{k}\) is found in the formula for the term: \(T_k = \binom{n}{k} a^{n-k} b^k\).
• The value of \(\binom{n}{k}\) is calculated using factorial, expressed as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Remembering how to calculate and use binomial coefficients can make tackling binomial expansions much easier.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition or subtraction). Expressions such as \(6x-3y\) in the context of a binomial like \((6x - 3y)^7\) often require structural manipulation when raising them to a power.

• To effectively handle algebraic expressions in binomial expansions, one must understand how each part interacts within the larger framework of the expression.
• Recognizing that each term consists of a coefficient and variable part — as seen with \(a = 6x\) and \(b = -3y\)

When these parts are raised to different powers, they form the complete expression in a binomial expansion. This enables finding particular terms efficiently in the expression, like the third term in the previous problem example.