(x+4)(x^2-4x+16)-(64-x^3)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (x+4)(x^2-4x+16)-(64-x^3)

This article will explore how to simplify the expression (x+4)(x^2-4x+16)-(64-x^3). We'll delve into the process, step-by-step, and explain the reasoning behind each action.

Understanding the Expression

At first glance, the expression might seem complex. However, it's built from recognizable patterns that we can exploit for simplification.

• The first part (x+4)(x^2-4x+16): This resembles the pattern of a sum of cubes, where (a+b)(a^2-ab+b^2) = a^3 + b^3.
• The second part (64-x^3): This is a difference of cubes pattern, where (a^3 - b^3) = (a-b)(a^2 + ab + b^2).

Simplifying using the Patterns

Let's apply these patterns to our expression:

1. Identify 'a' and 'b':

   • In the first part, a = x and b = 4.
   • In the second part, a = 4 and b = x.

2. Apply the patterns:

   • The first part becomes: x^3 + 4^3 = x^3 + 64.
   • The second part becomes: (4-x)(4^2 + 4x + x^2) = (4-x)(16 + 4x + x^2).

3. Rewrite the expression: The complete expression now looks like this: x^3 + 64 - (4-x)(16 + 4x + x^2).

4. Simplify further:

   • Distribute the negative sign: x^3 + 64 - 64 - 4x^2 - x^3 + 16x + 4x^2.
   • Combine like terms: 16x.

Final Result

By applying the patterns of sum and difference of cubes and simplifying the expression, we arrive at the final answer: 16x.