Final answer:
Factoring 2x^2 - x - 1 using a geometric model involves visually representing the polynomial with an area model, rearranging the model into a rectangle, and determining the rectangle's dimensions to find the factors, resulting in (2x + 1)(x - 1).
Step-by-step explanation:
To factor 2x2 - x - 1 using a geometric model, we will first model the polynomial as a product of areas. This approach involves creating a visual representation, often using algebra tiles or a grid, where we manipulate the placement of tiles to represent the terms of the polynomial and rearrange them to form a rectangle. The sides of this rectangle will then correspond to the factors of the polynomial.
Step 1: Model the polynomial by creating a grid with areas that represent 2x2, -x, and -1.
Step 2: Add a zero-pair tile if needed to rearrange the existing tiles into a rectangular shape.
Step 3: Once the tiles form a rectangle, drag tiles to represent the factors of the polynomial. Each dimension of the rectangle symbolizes a factor, such as (2x + a) and (x + b), where a and b are values that satisfy the equation when multiplied.
By modeling and rearranging the tiles, we can visualize the factors of the given quadratic equation and gain a deeper understanding of the process of factoring. The goal is to identify a pair of numbers that multiply to give the constant term (-1 in this case) and add to give the coefficient of the x term (-1). In the case of 2x2 - x - 1, those numbers would be 2x and -1 for the first factor, and x and 1 for the second factor, resulting in the factors (2x + 1)(x - 1).