Math Problem Statement

Expand (a + b)^4 using Pascal's Triangle.

Solution

To expand (a+b)4(a + b)^4 using Pascal's Triangle, we use the fourth row of Pascal's Triangle since the exponent is 44.

Pascal's Triangle:

  1. The first few rows of Pascal’s Triangle are as follows:
    • Row 0: 11
    • Row 1: 1,11, 1
    • Row 2: 1,2,11, 2, 1
    • Row 3: 1,3,3,11, 3, 3, 1
    • Row 4: 1,4,6,4,11, 4, 6, 4, 1

The coefficients for the expansion come from Row 4: 1,4,6,4,11, 4, 6, 4, 1.

Expansion:

(a+b)4=1a4+4a3b+6a2b2+4ab3+1b4(a + b)^4 = 1 \cdot a^4 + 4 \cdot a^3b + 6 \cdot a^2b^2 + 4 \cdot ab^3 + 1 \cdot b^4

Simplified Result:

(a+b)4=a4+4a3b+6a2b2+4ab3+b4(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4

Let me know if you'd like further clarification or a breakdown of the steps!


Relative Questions:

  1. How is Pascal's Triangle constructed for any row?
  2. What is the binomial theorem, and how does it relate to Pascal’s Triangle?
  3. How would you expand (ab)4(a - b)^4 using Pascal’s Triangle?
  4. Can you derive the coefficients for any power nn without constructing the full Pascal's Triangle?
  5. What applications does Pascal's Triangle have beyond binomial expansions?

Tip:

To quickly find a specific row in Pascal's Triangle, remember that the nn-th row corresponds to the coefficients of (a+b)n(a + b)^n and is based on combinations C(n,k)C(n, k).

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Math Problem Analysis

Mathematical Concepts

Binomial Expansion
Pascal's Triangle
Combinatorics

Formulas

(a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n

Theorems

Binomial Theorem
Pascal's Triangle Properties

Suitable Grade Level

Grades 7-9