Question

Write the area of the rectangle at right as a sum, and as a product

Answer 1

The area of the rectangle can be expressed as a sum and a product:

Sum:
`\(-1x - 3y + 5 + 2x^2 + 6xy - 10x\)`

Product:
`\(x(-9 + 2x) + y(-3 + 6x) + 5\)`

Let's denote the sides of the rectangle as a and b, where a is the length and b is the width.

The rectangle is divided into six parts:

Upper parts:

1. -1x

2. -3y

3. 5

Lower parts:

4.
`\(2x^2\)`

5. 6xy

6. -10x

The area A of the rectangle is given by the product of its length and width:

`\[ A = a \cdot b \]`

To express A as a sum, we add the areas of the individual parts:

`\[ A = (-1x) + (-3y) + 5 + 2x^2 + 6xy + (-10x) \]`

Now, to express A as a product, we factor out common terms:

`\[ A = x(-1 + 2x - 10) + y(-3 + 6x) + 5 \]`

So, the area of the rectangle can be expressed as a sum and a product:

Sum:
`\(-1x - 3y + 5 + 2x^2 + 6xy - 10x\)`

Product:
`\(x(-9 + 2x) + y(-3 + 6x) + 5\)`