Question

Use an area model to determine the product of n-3p

a) n^2 - 9p^2
b) n^2 - 6np + 9p^2
c) n^2 + 6np + 9p^2
d) n^2 - 6np - 9p^2

Answer 1

By using an area model, we multiply two binomials (n - 3p) and find that the two middle terms cancel each other out, leaving us with a product of n^2 - 9p^2, which is option (a).

Step-by-step explanation:

To use an area model to determine the product of (n - 3p), we must first understand that an area model represents multiplication visually by creating rectangles whose sides represent the different parts of the equation being multiplied. For the expression (n - 3p), we are essentially multiplying two binomials: (n + (-3p)) × (n + (-3p)). This is a case of the difference of squares because n and 3p are both squared, and their middle terms cancel each other out.

To find the product using the area model:

1. Draw a two-by-two grid.
2. Write n along the top of the two columns and -3p along the side of the two rows.
3. Fill in the four areas by multiplying the side lengths.
4. The top-left area is n times n, which gives n².
5. The top-right and bottom-left areas are both n times -3p, giving -3np each, but these will cancel out because they are additive inverses of each other.
6. The bottom-right area is -3p times -3p, which gives 9p².

Adding these together and combining like terms, we find the final answer to be n² - 9p², representing the area of the entire model. This corresponds to option (a).