(x%5E2%2B3x%2B9)-(x%5E3%2B3).jpeg "(x-3)(x^2+3x+9)-(x^3+3)")
Simplifying the Expression: (x-3)(x^2+3x+9)-(x^3+3)
This article explores the simplification of the algebraic expression: (x-3)(x^2+3x+9)-(x^3+3).
Understanding the Expression
The expression involves two distinct parts:
- (x-3)(x^2+3x+9): This is a product of two binomials. The second binomial (x^2+3x+9) is a special form known as the sum of cubes.
- (x^3+3): This is a simple binomial, representing a sum of two terms.
Applying the Sum of Cubes Formula
The sum of cubes formula states: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In our expression, we can apply this formula to the second binomial:
- a = x
- b = 3
Therefore, (x^2 + 3x + 9) = (x + 3)(x^2 - 3x + 9)
Expanding the Expression
Let's expand the entire expression:
(x-3)(x^2+3x+9)-(x^3+3) = (x-3)(x+3)(x^2-3x+9) - (x^3+3)
Now, we can apply the difference of squares formula: a^2 - b^2 = (a+b)(a-b) to the first part:
= [(x^2 - 9)(x^2 - 3x + 9)] - (x^3+3)
Expanding further:
= (x^4 - 3x^3 + 9x^2 - 9x^2 + 27x - 81) - (x^3 + 3)
Combining like terms:
= x^4 - 4x^3 + 27x - 84
Final Simplified Expression
Therefore, the simplified form of the expression (x-3)(x^2+3x+9)-(x^3+3) is x^4 - 4x^3 + 27x - 84.