https://file.aiquickdraw.com/mathbot/file/17261449130785jf76o4h.jpg

The image shows a question related to the **Pascal's Triangle** (Pascalsches Dreieck in German). The problem asks to fully expand the expression \((4x + 1)^4\) using the binomial theorem with the help of Pascal's Triangle.

### Binomial Expansion Formula:
The binomial expansion of \((a + b)^n\) is given by:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
where \(\binom{n}{k}\) are binomial coefficients, which can be found using Pascal's Triangle.

### Step-by-Step Expansion of \((4x + 1)^4\):

1. **Identify the values:**
   - \(a = 4x\)
   - \(b = 1\)
   - \(n = 4\)

2. **Pascal's Triangle for \(n = 4\):**
   The row for \(n = 4\) in Pascal's Triangle is:
   \[
   1, 4, 6, 4, 1
   \]
   These are the binomial coefficients.

3. **Apply the Binomial Theorem:**
   Using the binomial expansion formula, the expression becomes:
   \[
   (4x + 1)^4 = 1 \cdot (4x)^4 + 4 \cdot (4x)^3 \cdot 1 + 6 \cdot (4x)^2 \cdot 1^2 + 4 \cdot (4x)^1 \cdot 1^3 + 1 \cdot 1^4
   \]

4. **Simplify each term:**
   - \(1 \cdot (4x)^4 = 256x^4\)
   - \(4 \cdot (4x)^3 \cdot 1 = 4 \cdot 64x^3 = 256x^3\)
   - \(6 \cdot (4x)^2 \cdot 1^2 = 6 \cdot 16x^2 = 96x^2\)
   - \(4 \cdot (4x)^1 \cdot 1^3 = 4 \cdot 4x = 16x\)
   - \(1 \cdot 1^4 = 1\)

5. **Final expansion:**
   \[
   (4x + 1)^4 = 256x^4 + 256x^3 + 96x^2 + 16x + 1
   \]

So, the simplified form of \((4x + 1)^4\) is:
\[
\boxed{256x^4 + 256x^3 + 96x^2 + 16x + 1}
\]

---

Do you need further details or have any questions?  
Here are five related questions for further practice:

1. What is the expansion of \((3x + 2)^3\) using Pascal's Triangle?
2. How does Pascal's Triangle help in binomial expansions?
3. What is the general formula for the binomial expansion of \((a + b)^n\)?
4. Can you explain the role of binomial coefficients in the expansion?
5. How would you expand \((2x - 5)^5\) using Pascal's Triangle?

**Tip:** Pascal's Triangle can also be used to solve combinatorics problems, such as counting paths in a grid or selecting groups.

Erweitern Sie den folgenden Ausdruck vollständig mit Hilfe des Pascalschen Dreiecks: (4x + 1)^4

Learn how to expand the expression (4x + 1)^4 using the binomial theorem and Pascal's Triangle. This step-by-step solution involves identifying coefficients and applying algebraic techniques, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation