Chapter 8.7, Problem 3GP

To determine

To write: The factor form of a polynomial that represents the area of the colored glass if the clear squares have a side of length of 6 inches shown in given figure.

Answer to Problem 3GP

The area of colored glass in factored form of a polynomial is $A=x^2-144=\left(x-12\right)\left(x+12\right)$ .

Explanation of Solution

Given information:

The clear squares are shown in Figure here.

  

Concept used:

The difference of two squares can be factored as $a^2-b^2=\left(a-b\right)\left(a+b\right)$ .

Calculations:

Let each side of largest square is x as shown in Figure-1 here. Let the whole area of colored glass is denoted by A.

  

Clearly, the area of colored glass is $A=x^2-4\times \text{area}\text{of}\text{one}\text{corner}\text{quare}$

  $A=x^2-4\times {\left(\text{6}\right)}^2\text{inc}{\text{h}}^{\text{2}}$

  $\Rightarrow A=x^2-144$

  $\Rightarrow A=x^2-{\left(12\right)}^2$

The polynomial can be written as $A=x^2-144$ , i.e., is a difference of two squares.

Therefore, the polynomial $A=x^2-144$ can be factorized as

  $A=x^2-{\left(12\right)}^2=\left(x-12\right)\left(x+12\right)$

Thus, the area of colored glass in factored form of a polynomial is $A=x^2-144=\left(x-12\right)\left(x+12\right)$ .

Conclusion:

The area of colored glass in factored form of a polynomial is $A=x^2-144=\left(x-12\right)\left(x+12\right)$ .