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

Jasonbradley

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Jun 16, 2024

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Expanding the Expression: (x+2)(x^2 - 2x + 4)

This expression represents the multiplication of a binomial and a trinomial. To simplify it, we can use the distributive property. This means we multiply each term of the binomial (x + 2) by each term of the trinomial (x^2 - 2x + 4).

Here's how it works:

1. Multiply x by each term of the trinomial:

   • x * x^2 = x^3
   • x * (-2x) = -2x^2
   • x * 4 = 4x

2. Multiply 2 by each term of 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 + 8

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

Interesting Observation:

This expression is a special case of the sum of cubes factorization. Notice that the trinomial (x^2 - 2x + 4) is a result of squaring the first term of the binomial (x), changing the sign of the second term (2x), and squaring the second term (2). This pattern always leads to a simplified expression with just two terms:

(a + b)(a^2 - ab + b^2) = a^3 + b^3

In our case, a = x and b = 2.