Question

Express the area of the entire rectangle.
Your answer should be a polynomial in standard form.
An area model for a rectangle that has a height of x plus three and a width of x plus seven. The rectangle is broken into four rectangles to isolate each term in the height and the width. The top left rectangle has a height of x and a width of x. The top right rectangle has a height of x and width of seven. The bottom left rectangle has a height of three and a width of x. The bottom right rectangle has a height of three and a width of seven.

Answer 1

To find the area of the rectangle, we need to multiply the height and the width together.

Given:
- Height = \( x + 3 \)
- Width = \( x + 7 \)

The area \( A \) can be expressed as:

\[
A = (x + 3)(x + 7)
\]

Now we will expand this expression using the distributive property (FOIL method):

\[
A = x \cdot x + x \cdot 7 + 3 \cdot x + 3 \cdot 7
\]

Calculating each term:

1. \( x \cdot x = x^2 \)
2. \( x \cdot 7 = 7x \)
3. \( 3 \cdot x = 3x \)
4. \( 3 \cdot 7 = 21 \)

Now, combine all the terms:

\[
A = x^2 + 7x + 3x + 21
\]

Combine the like terms:

\[
A = x^2 + (7x + 3x) + 21
\]
\[
A = x^2 + 10x + 21
\]

Thus, the area of the rectangle in standard polynomial form is:

\[
\boxed{x^2 + 10x + 21}
\]