(x^3+8)(x-1)/x^2-2x+4

3 min read Jun 17, 2024

Simplifying the Expression: (x^3 + 8)(x - 1) / (x^2 - 2x + 4)

This article will guide you through simplifying the algebraic expression: (x^3 + 8)(x - 1) / (x^2 - 2x + 4). We'll break down the steps and use key algebraic concepts to achieve a simplified form.

Recognizing Key Patterns

Let's start by recognizing some important patterns within the expression:

Sum of Cubes: The term (x^3 + 8) can be factored using the sum of cubes formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2). In this case, a = x and b = 2.
Perfect Square Trinomial: The term (x^2 - 2x + 4) resembles a perfect square trinomial. This pattern helps us simplify it further.

Applying the Formulas

Factoring the Sum of Cubes: (x^3 + 8) = (x + 2)(x^2 - 2x + 4)
Simplifying the Expression: Now, we can substitute the factored terms back into the original expression: [(x + 2)(x^2 - 2x + 4)(x - 1)] / (x^2 - 2x + 4)
Canceling Common Factors: Notice that (x^2 - 2x + 4) appears in both the numerator and denominator. We can cancel these common factors: (x + 2)(x - 1)

The Simplified Form

The simplified form of the expression (x^3 + 8)(x - 1) / (x^2 - 2x + 4) is (x + 2)(x - 1).

Conclusion

By applying the sum of cubes formula and recognizing the pattern of a perfect square trinomial, we were able to simplify the given expression. This example highlights how understanding algebraic patterns and formulas can lead to efficient simplification of complex expressions.