To find a polynomial that represents the area of the square given that one side of the square is represented by the polynomial [tex]\( 8x + 7 \)[/tex], let's proceed with the necessary steps.
1. Understand the problem: The area [tex]\( A \)[/tex] of a square is given by the square of its side length [tex]\( s \)[/tex]. If the side length is given by [tex]\( s = 8x + 7 \)[/tex], then the area [tex]\( A \)[/tex] can be calculated as [tex]\( A = s^2 \)[/tex].
2. Set up the expression for the area: We have
[tex]\[
A = (8x + 7)^2
\][/tex]
3. Expand the polynomial expression:
To find the simplified polynomial for the area, we need to expand the binomial [tex]\( (8x + 7)^2 \)[/tex].
[tex]\[
(8x + 7)^2 = (8x + 7)(8x + 7)
\][/tex]
4. Apply the distributive property (also known as the FOIL method for binomials):
[tex]\[
(8x + 7)(8x + 7) = 8x \cdot 8x + 8x \cdot 7 + 7 \cdot 8x + 7 \cdot 7
\][/tex]
5. Calculate each term:
[tex]\[
8x \cdot 8x = 64x^2
\][/tex]
[tex]\[
8x \cdot 7 = 56x
\][/tex]
[tex]\[
7 \cdot 8x = 56x
\][/tex]
[tex]\[
7 \cdot 7 = 49
\][/tex]
6. Combine like terms: Add all these terms together:
[tex]\[
64x^2 + 56x + 56x + 49
\][/tex]
Simplify by combining the like terms:
[tex]\[
64x^2 + 112x + 49
\][/tex]
Thus, the simplified polynomial that represents the area of the square is:
[tex]\[
A = 64x^2 + 112x + 49
\][/tex]
