Final answer:
Area models represent multiplication of algebraic expressions. For each expression, a rectangle or square can be used as an area model, with the dimensions of the expressions determining the sides of the shape. The area can be divided into smaller sections, and the equation can be written to show that the area as a product equals the area as a sum.
Step-by-step explanation:
Area models are used to represent multiplication of algebraic expressions. To sketch an area model for each of the given multiplication expressions, we can create a rectangle or square and divide it into smaller sections based on the dimensions of the expressions. Here are the area models for each expression:
- (r + 1)(x + 2): The area model can be a rectangle with sides of length r + 1 and x + 2. We can divide the rectangle into four sections: one with dimensions r × x, one with dimensions 1 × r, one with dimensions 1 × x, and one with dimensions 1 × 1. The equation is (r + 1)(x + 2) = r × x + r + x + 2.
- 3(2x + 5): The area model can be a rectangle with sides of length 3 and 2x + 5. We can divide the rectangle into two sections: one with dimensions 3 × 2x and one with dimensions 3 × 5. The equation is 3(2x + 5) = 6x + 15.
- (2x^3)(x + 2): The area model can be a rectangle with sides of length 2x^3 and x + 2. We can divide the rectangle into two sections: one with dimensions 2x^3 × x and one with dimensions 2x^3 × 2. The equation is (2x^3)(x + 2) = 2x^4 + 4x^3.
- (x - 1)(y - 1): The area model can be a rectangle with sides of length x - 1 and y - 1. We can divide the rectangle into four sections: one with dimensions x × y, one with dimensions -1 × x, one with dimensions -1 × y, and one with dimensions -1 × -1. The equation is (x - 1)(y - 1) = xy - x - y + 1.