%5E2-8(x%5E2-x)%2B12-%E5%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3.jpeg "(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)