Express the area of the entire rectangle.

Question

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:

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