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 five 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 five and a width of x. The bottom right rectangle has a height of five and a width of seven.

Answer 1

To find the area of the rectangle given the height \( (x + 5) \) and width \( (x + 7) \), we can express the area \( A \) as the product of the height and the width:

\[
A = (x + 5)(x + 7)
\]

Next, we will use the distributive property (also known as the FOIL method for binomials) to expand this expression:

1. **First**: Multiply the first terms:
\[
x \cdot x = x^2
\]

2. **Outer**: Multiply the outer terms:
\[
x \cdot 7 = 7x
\]

3. **Inner**: Multiply the inner terms:
\[
5 \cdot x = 5x
\]

4. **Last**: Multiply the last terms:
\[
5 \cdot 7 = 35
\]

Now, combine all these results together:

\[
A = x^2 + 7x + 5x + 35
\]

Combine the like terms (\( 7x \) and \( 5x \)):

\[
A = x^2 + 12x + 35
\]

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

\[
\boxed{x^2 + 12x + 35}
\]