(x%5E2-2x%2B1)%2F4%2B3x-x%5E2.jpeg "(x+2)(x^2-2x+1)/4+3x-x^2")
Simplifying the Expression (x+2)(x^2-2x+1)/4 + 3x - x^2
This article will guide you through simplifying the expression (x+2)(x^2-2x+1)/4 + 3x - x^2. We will break down the process step by step, making it easy to understand.
Step 1: Factor the Quadratic Expression
The expression (x^2-2x+1) is a perfect square trinomial. It can be factored as (x-1)^2.
Step 2: Substitute and Simplify
Substituting the factored expression, we get:
(x+2)(x-1)^2/4 + 3x - x^2
Now, let's expand the numerator:
(x+2)(x-1)(x-1)/4 + 3x - x^2
=(x^2 - x - 2x + 2)(x-1)/4 + 3x - x^2
=(x^2 - 3x + 2)(x-1)/4 + 3x - x^2
Step 3: Expand and Combine Like Terms
Expand the numerator and combine like terms:
(x^3 - 3x^2 + 2x - x^2 + 3x - 2)/4 + 3x - x^2
=(x^3 - 4x^2 + 5x - 2)/4 + 3x - x^2
=(1/4)x^3 - x^2 + (5/4)x - (1/2) + 3x - x^2
=(1/4)x^3 - 2x^2 + (17/4)x - (1/2)
Conclusion
Therefore, the simplified form of the expression (x+2)(x^2-2x+1)/4 + 3x - x^2 is (1/4)x^3 - 2x^2 + (17/4)x - (1/2).