Math Problem Statement
Expand (a + b)^4 using Pascal's Triangle.
Solution
To expand using Pascal's Triangle, we use the fourth row of Pascal's Triangle since the exponent is .
Pascal's Triangle:
- The first few rows of Pascal’s Triangle are as follows:
- Row 0:
- Row 1:
- Row 2:
- Row 3:
- Row 4:
The coefficients for the expansion come from Row 4: .
Expansion:
Simplified Result:
Let me know if you'd like further clarification or a breakdown of the steps!
Relative Questions:
- How is Pascal's Triangle constructed for any row?
- What is the binomial theorem, and how does it relate to Pascal’s Triangle?
- How would you expand using Pascal’s Triangle?
- Can you derive the coefficients for any power without constructing the full Pascal's Triangle?
- What applications does Pascal's Triangle have beyond binomial expansions?
Tip:
To quickly find a specific row in Pascal's Triangle, remember that the -th row corresponds to the coefficients of and is based on combinations .
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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