Problem 36 - College Algebra

Question

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The tenth term of \((x-1)^{12}\)

Step-by-Step Solution

Answer

The tenth term is \(-220x^3\).

Step 1: Identifying Binomial Coefficients

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = x\), \(b = -1\), and \(n = 12\). We are asked to find the tenth term, which corresponds to \(k=9\) since binomial coefficients are zero-based.

Step 2: Calculating the Binomial Coefficient

We need to calculate \(\binom{12}{9}\) to find the coefficient for the tenth term. Using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), we get:\[\binom{12}{9} = \frac{12!}{9!(12-9)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220\]

Step 3: Constructing the Tenth Term

Using the binomial theorem, the tenth term is given by the expression:\[\binom{12}{9} (x)^{12-9}(-1)^9 = 220 \times x^3 \times (-1)\]Calculate the sign and simplify:\[= -220x^3\]

Step 4: Stating the Result

Thus, the tenth term of the expansion of \((x-1)^{12}\) is \(-220x^3\).

Key Concepts

Binomial Coefficients

The binomial coefficients are a fundamental part of the binomial theorem, which helps us expand expressions of the form \((a + b)^n\). These coefficients can be understood as the numerical factors in each term of the expanded polynomial. They are represented by \(\binom{n}{k}\), which is pronounced as "n choose k".
To calculate the binomial coefficient \(\binom{n}{k}\), the formula is \(\frac{n!}{k!(n-k)!}\). The exclamation point denotes factorial, which is the product of all positive integers up to that number. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
This coefficient tells us how many ways we can choose \(k\) elements from a total of \(n\) elements.

• For our specific case, the tenth term is associated with \(k=9\) because binomial coefficients use a zero-based index.
• Thus, \(\binom{12}{9} = 220\), indicating how the contents of the binomial \((x - 1)^{12}\) arrange themselves to produce this specific term.

Polynomial Expansion

Polynomial expansion is the process of multiplying out a polynomial expression, often using rules like the binomial theorem to simplify the process. In algebra, polynomial expansions help us understand how expressions combine to form more complex equations.
The binomial theorem allows us to efficiently expand expressions of the form \((a + b)^n\) without having to manually multiply the expression by itself \(n\) times.
Using the formula:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Each term in the expansion is given by a distinct \(k\), forming a complete set of terms when \(k\) runs from 0 to \(n\).

For example, in our exercise, the expansion of \((x-1)^{12}\) helps us determine the specific characteristics of the term we seek, the tenth term. Instead of fully expanding, we utilize the theorem to go directly to that term by correctly choosing the appropriate coefficients and powers of \(x\) and \(-1\).

Algebraic Expressions

Algebraic expressions use variables along with numbers and are fundamental to all algebraic mathematics. These expressions allow us to simplify complex equations and solve for unknown quantities.
In our context, \((x-1)^{12}\) is an algebraic expression where variables and constants are combined following algebraic operations like addition and multiplication. Understanding and manipulating these expressions are vital in solving algebra problems effectively.

• The structure of an algebraic expression includes a relationship between the coefficients and powers of variables.
• A clear understanding of each component helps us in extracting specific terms, such as the tenth term in a polynomial expansion.

When we identified that the tenth term corresponds to \(-220x^3\), we used our understanding of algebraic expressions to simplify and determine both the coefficient and power of \(x\). This approach illustrates how algebra supports problem-solving by breaking down larger, more complex expressions into manageable parts.