%5E3-expand.jpeg "(x+1)^3 Expand")
Expanding (x + 1)³
The expression (x + 1)³ represents the cube of the binomial (x + 1). Expanding this expression involves multiplying the binomial by itself three times. Here are two common methods to achieve this:
1. Using the Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ. In our case, a = x, b = 1, and n = 3. Applying the binomial theorem, we get:
(x + 1)³ = ³C₀x³(1)⁰ + ³C₁x²(1)¹ + ³C₂x¹(1)² + ³C₃x⁰(1)³
where ³C₀, ³C₁, ³C₂, and ³C₃ are binomial coefficients calculated using the formula:
nCr = n! / (r! * (n-r)!)
Calculating the coefficients, we have:
- ³C₀ = 1
- ³C₁ = 3
- ³C₂ = 3
- ³C₃ = 1
Substituting these values in the expanded form, we get:
(x + 1)³ = 1x³ + 3x² + 3x + 1
2. Direct Multiplication
Alternatively, we can directly multiply the binomial three times:
(x + 1)³ = (x + 1)(x + 1)(x + 1)
First, multiply the first two terms:
(x + 1)(x + 1) = x² + 2x + 1
Now, multiply this result by the remaining (x + 1):
(x² + 2x + 1)(x + 1) = x³ + 3x² + 3x + 1
Therefore, both methods lead to the same expansion:
(x + 1)³ = x³ + 3x² + 3x + 1
Conclusion
Expanding (x + 1)³ can be done using either the binomial theorem or direct multiplication. Both methods result in the same expanded form: x³ + 3x² + 3x + 1. Understanding these methods is essential for manipulating and simplifying algebraic expressions.