Chapter 8.4, Problem 45PPS

To determine

To determine the polynomial expression of the area of the given figure.

  

Answer to Problem 45PPS

  $x^2+4x+4$

Explanation of Solution

Given:

  

Formula used:

Area of Square = ${\text{(side)}}^2$

  ${(a-b)}^2=a^2-2\times a\times b+b^2$

  ${(a+b)}^2=a^2+2\times a\times b+b^2$

Calculation / Explanation:

First, consider the given figure and distributing the shapes as 1 and 2,

  

Now, considering shape 1 at the left, it is a square.

Side of the square $=x-1$

So, area of shape 1 is,

Area $\text{ = (}x-1{)}^2$

  $=x^2-2\times x\times 1+1^2\left[\because {(a-b)}^2=a^2-2\times a\times b+b^2\right]$

  $=x^2-2x+1$

Similarly, considering shape 2 at the right, it is also a square.

Side of square $=x+2$

So, area of shape 2 is,

Area $\text{ = (}x+2{)}^2$

  $=x^2+2\times x\times 2+2^2\left[\because {(a+b)}^2=a^2+2\times a\times b+b^2\right]$

  $=x^2+4x+4$

Now, as both the shapes are attached to each other. So, area of the total shape will be equal to sum of area of shape 1 and shape 2.

Total area = $\text{Sum of shape 1 + Sum of shape 2}$

  $\begin{array}{l}=(x^2-2x+1)+(x^2+4x+4)\\ =2x^2+2x+5\end{array}$

Conclusion:

So, the area of the shape at the right can be expressed as $x^2+4x+4$ .