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Expanding (x + a)³
The expansion of (x + a)³ is a fundamental concept in algebra, particularly when dealing with polynomial expressions. Understanding this expansion is crucial for various applications, including solving equations, simplifying expressions, and working with binomial theorems.
Understanding the Expansion
The expansion of (x + a)³ involves multiplying the entire expression by itself three times:
(x + a)³ = (x + a) * (x + a) * (x + a)
To achieve this expansion, we can employ the following methods:
1. Repeated Multiplication:
- Step 1: Expand (x + a) * (x + a) = x² + 2ax + a²
- Step 2: Multiply the result from Step 1 by (x + a): (x² + 2ax + a²) * (x + a) = x³ + 3ax² + 3a²x + a³
2. Using the Binomial Theorem:
The Binomial Theorem provides a general formula for expanding expressions of the form (x + a)ⁿ:
(x + a)ⁿ = ∑(n choose k) * x^(n-k) * a^k
where:
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
- k ranges from 0 to n
Applying this to (x + a)³, we get:
- (3 choose 0) * x³ * a⁰ + (3 choose 1) * x² * a¹ + (3 choose 2) * x¹ * a² + (3 choose 3) * x⁰ * a³
- This simplifies to: x³ + 3ax² + 3a²x + a³
The Resulting Expansion:
Therefore, the expansion of (x + a)³ is:
(x + a)³ = x³ + 3ax² + 3a²x + a³
Key Points:
- The expansion follows a pattern with coefficients that are the same as those found in Pascal's Triangle.
- The exponents of 'x' decrease from 3 to 0, while the exponents of 'a' increase from 0 to 3.
- This expansion is essential for working with cubic expressions and understanding higher-order polynomials.
By understanding the expansion of (x + a)³, you gain a valuable tool for manipulating algebraic expressions and simplifying equations.