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(x+2)^3-x(x+3)(x-3)-12x^2-8

2 min read Jun 16, 2024
(x+2)^3-x(x+3)(x-3)-12x^2-8

Simplifying the Expression (x+2)^3 - x(x+3)(x-3) - 12x^2 - 8

This article will guide you through simplifying the expression (x+2)^3 - x(x+3)(x-3) - 12x^2 - 8. We'll break down each step and explain the reasoning behind it.

Expanding the Expression

Let's start by expanding the expression step by step:

  1. (x+2)^3: This is a cube of a binomial. We can use the binomial theorem or simply multiply (x+2) by itself three times: (x+2)^3 = (x+2)(x+2)(x+2) = (x^2 + 4x + 4)(x+2) = x^3 + 6x^2 + 12x + 8

  2. x(x+3)(x-3): This is a product of three binomials. Notice that (x+3) and (x-3) form a difference of squares pattern: x(x+3)(x-3) = x(x^2 - 9) = x^3 - 9x

  3. -12x^2 - 8: This part remains unchanged for now.

Now, let's put everything together: (x+2)^3 - x(x+3)(x-3) - 12x^2 - 8 = (x^3 + 6x^2 + 12x + 8) - (x^3 - 9x) - 12x^2 - 8

Combining Like Terms

Finally, we can combine the like terms:

x^3 + 6x^2 + 12x + 8 - x^3 + 9x - 12x^2 - 8 = -6x^2 + 21x

Conclusion

Therefore, the simplified form of the expression (x+2)^3 - x(x+3)(x-3) - 12x^2 - 8 is -6x^2 + 21x.

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