(x+1)^3 Expand Formula

3 min read Jun 16, 2024

Expanding (x + 1)³

The expression (x + 1)³ represents the cube of the binomial (x + 1). To expand this expression, we can use the following methods:

1. Repeated Multiplication

The simplest method is to multiply (x + 1) by itself three times:

Step 1: (x + 1) * (x + 1) = x² + 2x + 1
Step 2: (x² + 2x + 1) * (x + 1) = x³ + 3x² + 3x + 1

Therefore, (x + 1)³ = x³ + 3x² + 3x + 1

2. Binomial Theorem

The Binomial Theorem provides a general formula for expanding any power of a binomial:

(a + b)ⁿ = ∑(n choose k) * a^(n-k) * b^k

Where:

n is the power to which the binomial is raised.
k ranges from 0 to n.
(n choose k) represents the binomial coefficient, which is calculated as n! / (k! * (n-k)!).

Applying this to (x + 1)³, we have:

n = 3
a = x
b = 1

Substituting these values into the Binomial Theorem, we get:

(x + 1)³ = (3 choose 0) * x³ * 1⁰ + (3 choose 1) * x² * 1¹ + (3 choose 2) * x¹ * 1² + (3 choose 3) * x⁰ * 1³

Calculating the binomial coefficients:

(3 choose 0) = 1
(3 choose 1) = 3
(3 choose 2) = 3
(3 choose 3) = 1

Substituting these values back into the equation:

(x + 1)³ = 1 * x³ * 1 + 3 * x² * 1 + 3 * x * 1 + 1 * 1 * 1

Simplifying the equation, we obtain:

**(x + 1)³ = x³ + 3x² + 3x + 1

3. Pascal's Triangle

Pascal's Triangle provides a visual representation of binomial coefficients. The numbers in each row correspond to the coefficients of the expanded binomial.

To expand (x + 1)³, we need the coefficients from the fourth row of Pascal's Triangle (remembering the top row is row 0):

1 3 3 1

These coefficients correspond to the terms in the expanded form of (x + 1)³:

(x + 1)³ = 1 * x³ + 3 * x² + 3 * x + 1 * 1

Simplifying the equation, we get:

**(x + 1)³ = x³ + 3x² + 3x + 1

Conclusion

All three methods demonstrate that the expanded form of (x + 1)³ is x³ + 3x² + 3x + 1. The method you choose depends on your preference and the complexity of the binomial expression.