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Exploring the Expansion of (x+1)(x+1)(x+1)
This article delves into the expansion of the expression (x+1)(x+1)(x+1), also known as (x+1)³. We'll discuss different methods for expanding the expression and explore its significance in algebra.
Method 1: Step-by-Step Expansion
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Expand the first two factors: (x+1)(x+1) = x² + 2x + 1
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Multiply the result by the remaining (x+1): (x² + 2x + 1)(x+1) = x³ + 2x² + x + x² + 2x + 1
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Combine like terms: x³ + 2x² + x + x² + 2x + 1 = x³ + 3x² + 3x + 1
Method 2: Using the Binomial Theorem
The Binomial Theorem provides a general formula for expanding expressions of the form (a + b)ⁿ. Applying it to (x+1)³, we get:
(x+1)³ = ¹C₀x³ + ¹C₁x² * 1¹ + ¹C₂x¹ * 1² + ¹C₃x⁰ * 1³
Where:
- ¹C₀, ¹C₁, ¹C₂, ¹C₃ represent binomial coefficients, which can be calculated using the formula nCr = n! / (r! * (n-r)!).
Evaluating the coefficients:
- ¹C₀ = 1
- ¹C₁ = 3
- ¹C₂ = 3
- ¹C₃ = 1
Substituting the values:
(x+1)³ = 1x³ + 3x² * 1 + 3x¹ * 1² + 1x⁰ * 1³ = x³ + 3x² + 3x + 1
Significance of the Result
The expanded form x³ + 3x² + 3x + 1 represents a cubic polynomial, a polynomial with a highest degree of 3. This form is important in various areas of mathematics:
- Algebra: It helps in understanding the behavior of cubic functions and their graphs.
- Calculus: The derivative of (x+1)³ is 3x² + 6x + 3, which finds applications in optimization problems.
- Physics: Cubic polynomials can model certain physical phenomena like the motion of objects under certain conditions.
Conclusion
The expansion of (x+1)(x+1)(x+1) reveals the power of algebraic manipulation and the significance of cubic polynomials. Whether using step-by-step expansion or the Binomial Theorem, the result provides valuable insights into the behavior and applications of this expression in diverse mathematical fields.