(x%5E2%2B3x%2B9)-simplify.jpeg "(x-3)(x^2+3x+9) Simplify")
Simplifying the Expression (x-3)(x^2+3x+9)
This expression represents the multiplication of a binomial and a trinomial. To simplify it, we can use the distributive property or recognize a specific pattern.
Understanding the Pattern
The expression (x^2+3x+9) is a special case of a sum of cubes factorization. It can be rewritten as:
(x^2 + 3x + 9) = (x + 3)(x^2 - 3x + 9)
This pattern allows us to simplify the entire expression easily.
Applying the Pattern
Let's substitute this back into our original expression:
(x-3)(x^2+3x+9) = (x-3)(x+3)(x^2-3x+9)
Now we have the product of two binomials and a trinomial. The first two binomials represent the difference of squares pattern:
(x-3)(x+3) = x^2 - 9
Therefore, our simplified expression becomes:
(x^2 - 9)(x^2 - 3x + 9)
Expanding (Optional)
If you want to fully expand the expression, you would multiply the remaining trinomial by the resulting binomial:
(x^2 - 9)(x^2 - 3x + 9) = x^4 - 3x^3 + 9x^2 - 9x^2 + 27x - 81
This simplifies to:
x^4 - 3x^3 + 27x - 81
Conclusion
By recognizing the sum of cubes pattern and applying the difference of squares pattern, we were able to simplify the expression (x-3)(x^2+3x+9) to (x^2 - 9)(x^2 - 3x + 9) or further expand it to x^4 - 3x^3 + 27x - 81.