(2x+3)^3 Answer

Jasonbradley

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Jun 16, 2024

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Expanding (2x + 3)³

Expanding a binomial raised to a power can be done using the Binomial Theorem or by repeated multiplication. Let's explore both methods for expanding (2x + 3)³.

Method 1: Binomial Theorem

The Binomial Theorem provides a formula for expanding (a + b)ⁿ:

(a + b)ⁿ = ₙC₀aⁿb⁰ + ₙC₁aⁿ⁻¹b¹ + ₙC₂aⁿ⁻²b² + ... + ₙCₙa⁰bⁿ

Where ₙCᵣ represents the binomial coefficient, calculated as:

ₙCᵣ = n! / (r! * (n-r)!)

Applying this to (2x + 3)³, we have:

• a = 2x
• b = 3
• n = 3

Therefore:

(2x + 3)³ = ³C₀(2x)³(3)⁰ + ³C₁(2x)²(3)¹ + ³C₂(2x)¹(3)² + ³C₃(2x)⁰(3)³

Calculating the binomial coefficients:

• ³C₀ = 3! / (0! * 3!) = 1
• ³C₁ = 3! / (1! * 2!) = 3
• ³C₂ = 3! / (2! * 1!) = 3
• ³C₃ = 3! / (3! * 0!) = 1

Substituting these values back into the equation:

(2x + 3)³ = 1(8x³)(1) + 3(4x²)(3) + 3(2x)(9) + 1(1)(27)

Simplifying:

(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 factors:

(2x + 3)(2x + 3) = 4x² + 6x + 6x + 9 = 4x² + 12x + 9

Now, multiply the result by the third factor:

(4x² + 12x + 9)(2x + 3) = 8x³ + 24x² + 18x + 12x² + 36x + 27

Combining like terms:

(2x + 3)³ = 8x³ + 36x² + 54x + 27

Conclusion

Both methods demonstrate that the expansion of (2x + 3)³ is 8x³ + 36x² + 54x + 27. You can choose the method that suits your preference and understanding best. Remember, practice makes perfect!