Problem 29 - Algebra and Trigonometry

Question

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{6}\) in the expansion of \((x+3)^{10}\)

Step-by-Step Solution

Answer

The coefficient of \( x^6 \) is 17010.

Step 1: Identify the General Term

The general term in the binomial expansion of \( (x + 3)^{10} \) is \[ T_k = \binom{10}{k} x^k (3)^{10-k} \]

Step 2: Set Up the Specific Term

We need to find the coefficient of \( x^6 \). Therefore, set \( k = 6 \) in the general term formula: \[ T_6 = \binom{10}{6} x^6 (3)^{10-6} \]

Step 3: Calculate the Binomial Coefficient

Calculate the binomial coefficient \( \binom{10}{6} \): \[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} \]

Step 4: Simplify the Binomial Coefficient

Simplify \( \binom{10}{6} \): \[ \binom{10}{6} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

Step 5: Complete the Term Calculation

Apply the binomial coefficient and simplify the term: \[ T_6 = 210 x^6 (3)^4 \] \[ T_6 = 210 x^6 (81) \]

Step 6: Combine the Coefficients

Combine the coefficients to find the coefficient of \( x^6 \): \[ 210 \times 81 = 17010 \]

Key Concepts

binomial expansion

The **Binomial Theorem** is a fundamental theorem in algebra that describes the algebraic expansion of powers of a binomial. A binomial is an algebraic expression with two terms, such as \( (x + y) \). According to the Binomial Theorem, any power of a binomial can be expanded into a sum of terms involving binomial coefficients. The theorem states that for any positive integer **n**, the expansion of \( (x + y)^n \) can be written as:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k} \]
Here, the terms involve binomial coefficients, which we will discuss in the next section. A key point to remember with binomial expansion is:
- **Each term in the expansion** has the form \( \binom{n}{k} x^{k} y^{n-k} \)
- **Binomial coefficients** are the numerical factors that multiply the powers of **x** and **y**. By systematically applying the theorem, finding specific terms or coefficients in the expansion becomes much easier. Let's dive deeper into these coefficients.

binomial coefficient

The **Binomial Coefficient** is a central part of the Binomial Theorem. It is denoted as \( \binom{n}{k} \) and represents the number of ways to choose **k** elements from a set of **n** elements without considering the order. It is sometimes read as 'n choose k'. The binomial coefficient is defined by the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, **n!** denotes the factorial of **n**, which is the product of all positive integers up to **n**. Factorials are pivotal when calculating binomial coefficients, as they account for all possible combinations.
- For example, to find \( \binom{10}{6} \), you set it up as:
\[ \binom{10}{6} = \frac{10!}{6!4!} \]
- Simplify the factorial calculations:
\[ \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]
Hence, \( \binom{10}{6} \) equals 210. Throughout binomial expansions, these coefficients give crucial numerical values that multiply the terms in the polynomial.

polynomial coefficients

In the context of the Binomial Theorem, **Polynomial Coefficients** refer to the numerical factors that precede the powers of the variables in polynomial expressions. These coefficients arise from the binomial coefficients and any additional factors in the expansion process. Let's refer back to our example \( (x + 3)^{10} \).
- The general term \( T_k \) in its expansion is given by:
\[ T_k = \binom{10}{k} x^{k} (3)^{10-k} \]
- To find the coefficient of **\( x^6 \)**, set **k = 6**:
\[ T_6 = \binom{10}{6} x^6 (3)^4 \]
Here, \( \binom{10}{6} = 210 \) and \( 3^4 = 81 \), producing the term:
\[ T_6 = 210 x^6 \cdot 81 = 17010 x^6 \]
The coefficient of \( x^6 \) is **17010**. These polynomial coefficients are crucial as they represent the exact magnitude of each term in the polynomial. They are derived from multiplying the binomial coefficients with the relevant powers of the constants inside the binomial.