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(x+2)^3 Simplify

2 min read Jun 16, 2024
(x+2)^3 Simplify

Simplifying (x + 2)³

The expression (x + 2)³ represents the cube of the binomial (x + 2). To simplify this, we can use the following methods:

Method 1: Expanding using the distributive property

This method involves multiplying the expression (x + 2) by itself three times.

  1. First Expansion: (x + 2)³ = (x + 2)(x + 2)(x + 2)

    = (x² + 4x + 4)(x + 2)

  2. Second Expansion: = x²(x + 2) + 4x(x + 2) + 4(x + 2) = x³ + 2x² + 4x² + 8x + 4x + 8

  3. Simplify: = x³ + 6x² + 12x + 8

Method 2: Using the binomial theorem

The binomial theorem provides a formula for expanding any binomial raised to a power. The formula for (x + 2)³ is:

(x + 2)³ = ³C₀ * x³ * 2⁰ + ³C₁ * x² * 2¹ + ³C₂ * x¹ * 2² + ³C₃ * x⁰ * 2³

Where:

  • ³C₀, ³C₁, ³C₂, ³C₃ are binomial coefficients calculated using the formula nCr = n! / (r! * (n-r)!).

Substituting the values:

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

Simplifying:

(x + 2)³ = x³ + 6x² + 12x + 8

Conclusion

Both methods lead to the same simplified form of (x + 2)³ which is x³ + 6x² + 12x + 8. The binomial theorem offers a faster solution for higher powers, while the distributive method is more intuitive and easily understandable.

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