(x+h)^3 expand

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29-03-2025

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Understanding how to expand algebraic expressions like (x + h)³ is fundamental in calculus and various branches of mathematics. This expression, a binomial cubed, appears frequently when calculating derivatives and understanding concepts like limits. This guide provides a clear, step-by-step explanation of how to expand (x + h)³ and offers different approaches to solve this problem.

Method 1: Repeated Application of the Distributive Property (FOIL)

This method involves applying the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last) repeatedly.

1. First Expansion: Begin by expanding (x + h)²: (x + h)² = (x + h)(x + h) = x² + 2xh + h²

2. Second Expansion: Now, multiply the result by (x + h): (x + h)(x² + 2xh + h²) = x(x² + 2xh + h²) + h(x² + 2xh + h²)

3. Distributive Property: Distribute the 'x' and 'h' to each term within the parentheses: x³ + 2x²h + xh² + x²h + 2xh² + h³

4. Combine Like Terms: Simplify by combining like terms: x³ + 3x²h + 3xh² + h³

Therefore, the expansion of (x + h)³ is x³ + 3x²h + 3xh² + h³.

Method 2: Using the Binomial Theorem

The binomial theorem provides a more general and efficient way to expand expressions of the form (a + b)ⁿ. For (x + h)³, we have a = x, b = h, and n = 3.

The binomial theorem states: (a + b)ⁿ = Σ (nCk) * aⁿ⁻ᵏ * bᵏ where k goes from 0 to n, and nCk represents the binomial coefficient "n choose k," calculated as n! / (k! * (n-k)!).

1. Apply the Binomial Theorem: Let's apply this to (x + h)³: (x + h)³ = (3C0)x³h⁰ + (3C1)x²h¹ + (3C2)x¹h² + (3C3)x⁰h³

2. Calculate Binomial Coefficients:

   • 3C0 = 1
   • 3C1 = 3
   • 3C2 = 3
   • 3C3 = 1

3. Substitute and Simplify: Substitute the binomial coefficients back into the equation: (x + h)³ = 1x³1 + 3x²h + 3xh² + 11h³ = x³ + 3x²h + 3xh² + h³

Again, the expansion of (x + h)³ is x³ + 3x²h + 3xh² + h³.

Method 3: Pascal's Triangle

Pascal's Triangle is a visual tool that helps determine the binomial coefficients. Each number in the triangle is the sum of the two numbers directly above it.

            1
           1 1
          1 2 1
         1 3 3 1
        1 4 6 4 1
       1 5 10 10 5 1
      ...and so on

The fourth row (1 3 3 1) represents the coefficients for (x + h)³. These coefficients are then used in the same way as in Method 2:

(x + h)³ = 1x³ + 3x²h + 3xh² + 1h³ = x³ + 3x²h + 3xh² + h³

Conclusion

Expanding (x + h)³ results in x³ + 3x²h + 3xh² + h³, regardless of the method used. Understanding these different methods provides valuable tools for tackling more complex algebraic expansions and further mathematical concepts. Choosing the method that you find easiest and most intuitive is key to mastering this fundamental skill. Remember that each method will yield the same result – the expanded form of (x+h)³.