https://file.aiquickdraw.com/mathbot/file/1729045399999uwnc2cby.jpg

Find the third term. Also what’s the difference between term and derivative simple explanation please and thanks

The expression provided is \((x + 5)^5\), and you have asked to find the third term of its expansion. This involves using the **Binomial Theorem**, which is used to expand expressions of the form \((a + b)^n\).

The Binomial Theorem states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
Where \(\binom{n}{k}\) is the binomial coefficient, calculated as:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

For \((x + 5)^5\), we have \(a = x\), \(b = 5\), and \(n = 5\).

The general term in the expansion is:
\[
T_k = \binom{5}{k} x^{5-k} \cdot 5^k
\]
We need to find the third term, which corresponds to \(k = 2\) (because the first term is for \(k = 0\)).

### Steps:
1. **Binomial Coefficient:**
   \[
   \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
   \]
2. **Power of \(x\):**
   \[
   x^{5-2} = x^3
   \]
3. **Power of \(5\):**
   \[
   5^2 = 25
   \]
4. **Putting it all together:**
   \[
   T_3 = 10 \cdot x^3 \cdot 25 = 250x^3
   \]

Thus, the **third term** in the expansion of \((x + 5)^5\) is:
\[
250x^3
\]

### Explanation of Terms vs Derivatives:
- **Term**: A term in an expansion or equation is one of the individual parts that are added or subtracted together (e.g., \(250x^3\) is a term in the expansion).
- **Derivative**: A derivative measures how a function changes as its input changes. It is a fundamental concept in calculus, representing the rate of change or slope of a function at any point.

Would you like more details on this? Here are some questions for further exploration:

1. What is the full expansion of \((x + 5)^5\)?
2. How is the binomial theorem applied to larger powers?
3. What does the fourth term of the expansion look like?
4. How do you calculate the derivative of a binomial expression like \((x + 5)^5\)?
5. What are some applications of binomial expansions in real-world scenarios?

**Tip**: When expanding binomials, use the binomial theorem to simplify large powers instead of expanding step by step manually.

Learn how to find the third term in the expansion of (x + 5)^5 using the binomial theorem. This explanation includes step-by-step calculation of the binomial coefficient and powers. It also covers the difference between a term in algebra and a derivative in calculus. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation