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

Jasonbradley

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Jun 17, 2024

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

This article will guide you through the process of factoring and simplifying the expression (x-4)^2(x+4)-(x-4)(x+4)^2+3(x^2-16).

Recognizing the Pattern

The expression contains repeated terms and the difference of squares pattern. Let's break it down:

• (x-4)^2(x+4): This term has (x-4) as a common factor.
• -(x-4)(x+4)^2: This term also has (x-4) as a common factor.
• 3(x^2-16): This term represents the difference of squares: (x+4)(x-4)

Factoring the Expression

1. Factor out the common factor (x-4):

   (x-4)^2(x+4) - (x-4)(x+4)^2 + 3(x^2-16) = (x-4)[(x-4)(x+4)-(x+4)^2 + 3(x+4)(x-4)]

2. Factor out the common factor (x+4) from the remaining expression:

   (x-4)[(x-4)(x+4)-(x+4)^2 + 3(x+4)(x-4)] = (x-4)(x+4)[(x-4)-(x+4) + 3(x-4)]

3. Simplify the expression within the brackets:

   (x-4)(x+4)[(x-4)-(x+4) + 3(x-4)] = (x-4)(x+4)[3x - 12]

4. Factor out the common factor 3 from the last term:

   (x-4)(x+4)[3x - 12] = 3(x-4)(x+4)(x-4)

Final Simplified Expression

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

This process demonstrates how recognizing patterns and factoring can significantly simplify complex algebraic expressions.