(x+5)(x^2-5x+25)-(x+5)^3

Jasonbradley

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Jun 17, 2024

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Simplifying the Expression (x+5)(x^2-5x+25)-(x+5)^3

This article will guide you through simplifying the given expression: (x+5)(x^2-5x+25)-(x+5)^3.

Understanding the Concepts

Before we begin simplifying, let's review some key algebraic concepts:

• Difference of Cubes: The formula for the difference of cubes is: a^3 - b^3 = (a-b)(a^2 + ab + b^2)
• Binomial Expansion: For a binomial raised to a power, we can use the binomial theorem or simply expand it out using the distributive property.

Simplifying the Expression

1. Recognize the Difference of Cubes: Observe that the first part of the expression, (x+5)(x^2-5x+25), resembles the expansion of the difference of cubes. We can rewrite it as:

   (x+5)(x^2-5x+25) = x^3 + 5^3 = x^3 + 125

2. Expand the Cube: The second part of the expression, (x+5)^3, can be expanded using the binomial theorem or simply by multiplying it out:

   (x+5)^3 = (x+5)(x+5)(x+5) = (x^2 + 10x + 25)(x+5) = x^3 + 15x^2 + 75x + 125

3. Combine the Terms: Now we can combine the simplified terms from steps 1 and 2:

   (x+5)(x^2-5x+25)-(x+5)^3 = (x^3 + 125) - (x^3 + 15x^2 + 75x + 125)

4. Simplify: Distribute the negative sign and combine like terms:

   x^3 + 125 - x^3 - 15x^2 - 75x - 125 = -15x^2 - 75x

Conclusion

Therefore, the simplified form of the expression (x+5)(x^2-5x+25)-(x+5)^3 is -15x^2 - 75x.