Math Problem Statement

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

Solution

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(4x + 1)^4 using the binomial theorem with the help of Pascal's Triangle.

Binomial Expansion Formula:

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

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

  1. Identify the values:

    • a=4xa = 4x
    • b=1b = 1
    • n=4n = 4
  2. Pascal's Triangle for n=4n = 4: The row for n=4n = 4 in Pascal's Triangle is: 1,4,6,4,11, 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(4x)4+4(4x)31+6(4x)212+4(4x)113+114(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(4x)4=256x41 \cdot (4x)^4 = 256x^4
    • 4(4x)31=464x3=256x34 \cdot (4x)^3 \cdot 1 = 4 \cdot 64x^3 = 256x^3
    • 6(4x)212=616x2=96x26 \cdot (4x)^2 \cdot 1^2 = 6 \cdot 16x^2 = 96x^2
    • 4(4x)113=44x=16x4 \cdot (4x)^1 \cdot 1^3 = 4 \cdot 4x = 16x
    • 114=11 \cdot 1^4 = 1
  5. Final expansion: (4x+1)4=256x4+256x3+96x2+16x+1(4x + 1)^4 = 256x^4 + 256x^3 + 96x^2 + 16x + 1

So, the simplified form of (4x+1)4(4x + 1)^4 is: 256x4+256x3+96x2+16x+1\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(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(a + b)^n?
  4. Can you explain the role of binomial coefficients in the expansion?
  5. How would you expand (2x5)5(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.

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Math Problem Analysis

Mathematical Concepts

Binomial Expansion
Algebra
Pascal's Triangle

Formulas

(a + b)^n = Σ [C(n, k) * a^(n-k) * b^k]
Pascal's Triangle coefficients

Theorems

Binomial Theorem

Suitable Grade Level

Grades 9-12