(x-2)(x^2+2x+4) Simplify

Jasonbradley

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Jun 17, 2024

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Simplifying the Expression (x-2)(x^2 + 2x + 4)

This expression involves multiplying a binomial (x-2) with a trinomial (x^2 + 2x + 4). We can simplify this using the distributive property (also known as FOIL method).

Understanding the Problem:

The expression represents the product of two factors:

• (x-2): This is a binomial, meaning it has two terms.
• (x^2 + 2x + 4): This is a trinomial, meaning it has three terms.

Simplifying the Expression:

To simplify this, we need to multiply each term in the binomial by each term in the trinomial:

1. Multiply x (from the binomial) with each term in the trinomial:

   • x * x^2 = x^3
   • x * 2x = 2x^2
   • x * 4 = 4x

2. Multiply -2 (from the binomial) with each term in the trinomial:

   • -2 * x^2 = -2x^2
   • -2 * 2x = -4x
   • -2 * 4 = -8

3. Combine all the terms:

   • x^3 + 2x^2 + 4x - 2x^2 - 4x - 8

4. Simplify by combining like terms:

   • x^3 + (2x^2 - 2x^2) + (4x - 4x) - 8
   • x^3 - 8

Therefore, the simplified expression of (x-2)(x^2 + 2x + 4) is x^3 - 8.

Important Note: The trinomial (x^2 + 2x + 4) is a special case known as the "sum of cubes" pattern. This pattern can be factored as: (a^3 + b^3) = (a + b)(a^2 - ab + b^2)

In this case, a = x and b = 2, making the simplification a straightforward application of this pattern.