(x%5E2-5x%2B25)-(x%2B3)%5E3%2B(x-2)(x%5E2%2B2x%2B4)-(x-1)%5E3.jpeg "(x+5)(x^2-5x+25)-(x+3)^3+(x-2)(x^2+2x+4)-(x-1)^3")
Simplifying a Complex Algebraic Expression
This article explores the simplification of the complex algebraic expression:
(x + 5)(x² - 5x + 25) - (x + 3)³ + (x - 2)(x² + 2x + 4) - (x - 1)³
We will employ several algebraic properties and techniques to achieve a simplified form.
Recognizing Special Products
The expression contains several special product forms:
- (x + 5)(x² - 5x + 25) This resembles the sum of cubes pattern: a³ + b³ = (a + b)(a² - ab + b²).
- (x + 3)³ This is a cube of a binomial: (a + b)³ = a³ + 3a²b + 3ab² + b³.
- (x - 2)(x² + 2x + 4) Another sum of cubes pattern: a³ - b³ = (a - b)(a² + ab + b²).
- (x - 1)³ Cube of a binomial: (a - b)³ = a³ - 3a²b + 3ab² - b³.
Applying the Special Product Patterns
Let's apply the special product patterns to expand the expression:
(x + 5)(x² - 5x + 25) - (x + 3)³ + (x - 2)(x² + 2x + 4) - (x - 1)³ =
(x³ + 5³) - (x³ + 3 * x² * 3 + 3 * x * 3² + 3³) + (x³ - 2³) - (x³ - 3 * x² * 1 + 3 * x * 1² - 1³) =
(x³ + 125) - (x³ + 9x² + 27x + 27) + (x³ - 8) - (x³ - 3x² + 3x - 1)
Simplifying by Combining Like Terms
Now, we can remove the parentheses and combine like terms:
x³ + 125 - x³ - 9x² - 27x - 27 + x³ - 8 - x³ + 3x² - 3x + 1 =
-6x² - 30x + 91
Final Simplified Expression
Therefore, the simplified form of the given algebraic expression is -6x² - 30x + 91.