Question

Use the Binomial Theorem to find the indicated coefficient or term. The 5 th term in the expansion of \((x+3)^{7}\)

Short answer

The_possess_X

Step-by-step solution

Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer n, t_is_the_coefficient_in_the_expansion_of_each_term

General term in Binomial Expansion

For the binomial expression we_n_that_the_general_term

Identify the given_term

Since_n=7, and_we_term_by_T

Set the general term for the_n term

The_equation_longer

Simplify to Find the Coefficient

Simplify beyond_coefficient

Key concepts

Binomial Expansion

The binomial expansion is a way to express the expansion of \((a+b)^{n}\). This theorem is very useful in algebra and has a wide range of applications such as in probability, calculus and number theory.

The binomial expansion formula is expressed as: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

Here, \ (a) \ and \ (b)\ are any numbers or variables, \ (n) \ is a non-negative integer, and \ (k) \ ranges from 0 to \ (n) \.

The notation \( \binom{n}{k} \) , read as 'n choose k,' refers to the binomial coefficient, which calculates the number of ways to choose \ (k) \ objects from \ (n) \ without regard to order. In simpler terms, it determines the coefficients of the terms in the binomial expansion.

Essentially, each term in the binomial expansion follows this pattern, and every term has a specific coefficient, which brings us to the next concept.

Coefficient

The coefficient in the binomial expansion is given by the binomial coefficient \ ( \binom{n}{k} ) \. This coefficient determines how many times each term in the binomial expansion will appear.

To find the coefficient of a particular term, use the formula \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

The factorial notation (!), means the product of all positive integers up to that number. For example, \ n! = n \times (n-1) \times (n-2) \times ... \times 1 \.

In the exercise given, we want to find the coefficient of the 5th term in the expansion of \ (x+3)^7 \.

Here's what we need to know:

• Identify \ n = 7 \.
• For the 5th term, \ k = 4 \ (since the first term corresponds to k=0)

Using the formula for the binomial coefficient: \[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1}= 35 \]
This will be multiplied by the remaining parts of the term to get the whole term's coefficient.

General Term

The general term in the binomial expansion helps to find any specific term without writing out the whole expansion. The general term is typically defined as: \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]

In our example of \ (x+3)^7 \, we are looking for the 5th term. Let's use the general term formula with \ n=7 \ and \ k=4 \.

Substituting in:

• \ a = x \
• \ b = 3 \
• \ n = 7 \
• \ k =4 \

\[ T_{5} = \binom{7}{4} x^{7-4} 3^4 \]

We already know from our coefficient calculation that \ \binom{7}{4} = 35 \. Now we simply finish by calculating:

• \ x^{7-4} = x^3 \
• \ 3^4 = 81 \

So, the fifth term becomes: \[ T_{5} = 35 \times x^3 \times 81 = 2835 x^3 \]
Thus, the 5th term in the expansion of \ (x+3)^7 \ is \ 2835 x^3 \.