Question

write a polynomial that represents the length of the rectangle. the width is x + 0.8 units. the area is 0.4x^3 + 0.22x^2 + 0.42x + 0.4 square units.

Explanation:

Step1: Recall area formula

The area formula of a rectangle is $A = lw$, where $A$ is the area, $l$ is the length and $w$ is the width. We need to find $l$, so $l=\frac{A}{w}$.

Step2: Set up the division

We have $A = 0.4x^{3}+0.22x^{2}+0.42x + 0.4$ and $w=x + 0.8$. So $l=\frac{0.4x^{3}+0.22x^{2}+0.42x + 0.4}{x + 0.8}$.

Step3: Perform polynomial long - division

Dividing $0.4x^{3}+0.22x^{2}+0.42x + 0.4$ by $x + 0.8$:
\[

$$\begin{align*} 0.4x^{3}+0.22x^{2}+0.42x + 0.4&=0.4x^{2}(x + 0.8)-0.1x^{2}+0.42x + 0.4\\ &=0.4x^{2}(x + 0.8)-0.1x(x + 0.8)+0.5x+0.4\\ &=0.4x^{2}(x + 0.8)-0.1x(x + 0.8)+0.5(x + 0.8) \end{align*}$$

\]
So $l = 0.4x^{2}-0.1x + 0.5$.

Answer:

$0.4x^{2}-0.1x + 0.5$