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: StepbyStep Expansion

Expand the first two factors: (x+1)(x+1) = x² + 2x + 1

Multiply the result by the remaining (x+1): (x² + 2x + 1)(x+1) = x³ + 2x² + x + x² + 2x + 1

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! * (nr)!).
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 stepbystep expansion or the Binomial Theorem, the result provides valuable insights into the behavior and applications of this expression in diverse mathematical fields.