3-answer.jpeg "(2x+3)3 Answer")
Expanding (2x + 3)³
The expression (2x + 3)³ represents the cube of the binomial (2x + 3). To expand this expression, we can use the following methods:
Method 1: Using the Binomial Theorem
The Binomial Theorem provides a general formula for expanding binomials raised to any power. For (2x + 3)³, it states:
(2x + 3)³ = ³C₀(2x)³ + ³C₁(2x)²(3) + ³C₂(2x)(3)² + ³C₃(3)³
Where ³Cᵣ represents the binomial coefficient, which can be calculated as:
³Cᵣ = 3! / (r! * (3-r)!)
Applying this formula:
- ³C₀ = 1
- ³C₁ = 3
- ³C₂ = 3
- ³C₃ = 1
Substituting these values and simplifying:
(2x + 3)³ = (1)(8x³) + (3)(4x²)(3) + (3)(2x)(9) + (1)(27)
Therefore, (2x + 3)³ = 8x³ + 36x² + 54x + 27
Method 2: Repeated Multiplication
We can expand (2x + 3)³ by multiplying (2x + 3) by itself three times:
(2x + 3)³ = (2x + 3)(2x + 3)(2x + 3)
First, multiply the first two binomials:
(2x + 3)(2x + 3) = 4x² + 6x + 6x + 9 = 4x² + 12x + 9
Then, multiply the result by the third binomial:
(4x² + 12x + 9)(2x + 3) = 8x³ + 24x² + 18x + 12x² + 36x + 27
Combining like terms:
Therefore, (2x + 3)³ = 8x³ + 36x² + 54x + 27
Both methods lead to the same result: (2x + 3)³ = 8x³ + 36x² + 54x + 27.