Question

Diagrams like the one in problem 3-87 are referred to as area models. Area models represent multiplication of algebraic expressions. For each multiplication expression, sketch an area model Label the dimensions and the area of each part. Then write an equation showing that the area as a product equals the area as a sum. a (x + 1)(x + 2) b. 3(2x + 5) c. (2x - 3)(x + 2) d. (x - 1)(y-1) e. --2y(y + 3) f.(-2+1)(3x + y - 4)

Answer 1

To sketch area models for various multiplication expressions, I will provide step-by-step explanations and examples for each expression, including the dimensions and areas of each part, as well as the corresponding equations.

Step-by-step explanation:

a. (x + 1)(x + 2)

To sketch the area model for (x + 1)(x + 2), you can draw a rectangle with dimensions x by x + 2. Divide the rectangle into four parts: a square with dimensions x by x, a square with dimensions 1 by 1, a rectangle with dimensions x by 2, and a rectangle with dimensions 1 by 2. The area of each part is x^2, 1, 2x, and 2. The equation is x^2 + 1 + 2x + 2 = (x + 1)(x + 2).

b. 3(2x + 5)

To sketch the area model for 3(2x + 5), you can draw a rectangle with dimensions 3 by (2x + 5). Divide the rectangle into two parts: a rectangle with dimensions 3 by 2x and a rectangle with dimensions 3 by 5. The areas are 6x and 15. The equation is 6x + 15 = 3(2x + 5).

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Answer 2

We have to sketch an area model for the expression:

`3(2x+5)`

We can write it mathematically as:

`3(2x+5)=3\cdot2x+3\cdot5=6x+15`

Then, the product of the sides 3 and (2x+5) are equal to the sum of 6x and 15.

Answer: 3(2x+5) = 6x+15