(x^2-x)^2-8(x^2-x)+12 因数分解

2 min read Jun 17, 2024
(x^2-x)^2-8(x^2-x)+12 因数分解

Factoring (x^2 - x)^2 - 8(x^2 - x) + 12

This problem involves factoring a quadratic expression, but with a twist. Let's break it down step by step.

Recognizing the Pattern

Notice that the expression has a repeated term: (x^2 - x). This is our key to simplifying the problem.

Substitution

Let's substitute y = (x^2 - x). Now our expression becomes:

y^2 - 8y + 12

This is a much more familiar quadratic expression!

Factoring the Quadratic

We can factor this quadratic expression easily:

(y - 6)(y - 2)

Back-Substitution

Now, we substitute back our original expression for y:

((x^2 - x) - 6)((x^2 - x) - 2)

Final Simplification

Let's simplify further:

(x^2 - x - 6)(x^2 - x - 2)

We can factor these two quadratics:

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

Final Answer

Therefore, the factored form of (x^2 - x)^2 - 8(x^2 - x) + 12 is:

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