(x-1)3-(x+2)(x2-2x+4)+3(x+4)(x-4)

3 min read Jun 17, 2024

Simplifying the Expression (x-1)³-(x+2)(x²-2x+4)+3(x+4)(x-4)

This article will guide you through the simplification of the algebraic expression (x-1)³-(x+2)(x²-2x+4)+3(x+4)(x-4). We will break down the steps and explain the underlying concepts.

Step 1: Expanding the Cubes and Products

Firstly, let's expand each part of the expression:

(x-1)³: This is a cube of a binomial. We can use the formula (a-b)³ = a³ - 3a²b + 3ab² - b³ to expand it.
- (x-1)³ = x³ - 3x²(1) + 3x(1)² - 1³ = x³ - 3x² + 3x - 1
(x+2)(x²-2x+4): This is a product of a binomial and a trinomial. We can use the distributive property to expand it.
- (x+2)(x²-2x+4) = x(x²-2x+4) + 2(x²-2x+4) = x³ - 2x² + 4x + 2x² - 4x + 8 = x³ + 8
3(x+4)(x-4): This is a product of a constant, a binomial, and its conjugate. The product of a binomial and its conjugate simplifies to the difference of squares.
- 3(x+4)(x-4) = 3(x² - 4²) = 3(x² - 16) = 3x² - 48

Step 2: Combining Like Terms

Now that we've expanded the expression, we can combine the like terms:

(x³ - 3x² + 3x - 1) - (x³ + 8) + (3x² - 48)

x³ terms: x³ - x³ = 0
x² terms: -3x² + 3x² = 0
x terms: 3x
Constant terms: -1 - 8 - 48 = -57

Step 3: The Simplified Expression

Finally, we can write the simplified expression:

(x-1)³-(x+2)(x²-2x+4)+3(x+4)(x-4) = 3x - 57

Therefore, the simplified form of the given expression is 3x - 57.

Key Concepts Used

Binomial Expansion: We used the formula for expanding the cube of a binomial.
Distributive Property: We used the distributive property to multiply binomials and trinomials.
Difference of Squares: We used the difference of squares pattern to simplify the product of a binomial and its conjugate.
Combining Like Terms: We combined terms with the same variable and exponent.