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Exploring the Expansion of (x+5)(x+5)(x+5)
This article will delve into the process of expanding the expression (x+5)(x+5)(x+5), also known as (x+5)³. We will explore different methods for simplifying this expression and understand the resulting polynomial.
Method 1: Step-by-Step Expansion
The most straightforward approach is to expand the expression step-by-step using the distributive property:
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Expand the first two factors: (x+5)(x+5) = x(x+5) + 5(x+5) = x² + 5x + 5x + 25 = x² + 10x + 25
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Multiply the result by the remaining factor: (x² + 10x + 25)(x+5) = x²(x+5) + 10x(x+5) + 25(x+5) = x³ + 5x² + 10x² + 50x + 25x + 125 = x³ + 15x² + 75x + 125
Therefore, the expanded form of (x+5)(x+5)(x+5) is x³ + 15x² + 75x + 125.
Method 2: Binomial Theorem
The Binomial Theorem provides a more efficient method for expanding expressions of the form (a+b)ⁿ. In our case, we have (x+5)³. The Binomial Theorem states:
(a+b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙ⁻¹abⁿ⁻¹ + bⁿ
where ⁿCᵣ represents the binomial coefficient, calculated as n! / (r! * (n-r)!).
Applying this to our expression:
(x+5)³ = x³ + ³C₁x²5¹ + ³C₂x¹5² + ³C₃x⁰5³
Calculating the binomial coefficients:
- ³C₁ = 3! / (1! * 2!) = 3
- ³C₂ = 3! / (2! * 1!) = 3
- ³C₃ = 3! / (3! * 0!) = 1
Substituting back into the equation:
(x+5)³ = x³ + 3x²5 + 3x5² + 5³ = x³ + 15x² + 75x + 125
Understanding the Result
The expansion of (x+5)³ results in a polynomial of degree 3, meaning the highest power of x is 3. It is a cubic polynomial with a constant term of 125 and coefficients 1, 15, and 75 for x³, x², and x respectively.
In conclusion, we have explored two different methods for expanding the expression (x+5)(x+5)(x+5), arriving at the same result of x³ + 15x² + 75x + 125. Understanding these methods allows for efficient manipulation of similar expressions and provides insights into the structure of polynomial functions.