Question

write a polynomial that represents the length of the rectangle. the width is x + 0.4 units. the area is 0.4x³ + 0.36x² + 0.48x + 0.16 square units. the length is □ units. (use integers or decimals for any numbers in the expression.)

Explanation:

Step1: Recall area formula for rectangle

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 of polynomials

We have $A = 0.4x^{3}+0.36x^{2}+0.48x + 0.16$ and $w=x + 0.4$. We perform polynomial long - division or synthetic division. Here we use polynomial long - division:
\[

$$\begin{align*} \frac{0.4x^{3}+0.36x^{2}+0.48x + 0.16}{x + 0.4}&=0.4x^{2}+\frac{0.36x^{2}+0.48x + 0.16-0.4x^{2}(x + 0.4)}{x + 0.4}\\ &=0.4x^{2}+\frac{0.36x^{2}+0.48x + 0.16-0.4x^{3}-0.16x^{2}}{x + 0.4}\\ &=0.4x^{2}+\frac{0.2x^{2}+0.48x + 0.16}{x + 0.4}\\ &=0.4x^{2}+0.2x+\frac{0.48x + 0.16-0.2x(x + 0.4)}{x + 0.4}\\ &=0.4x^{2}+0.2x+\frac{0.48x + 0.16-0.2x^{2}-0.08x}{x + 0.4}\\ &=0.4x^{2}+0.2x+\frac{0.4x + 0.16}{x + 0.4}\\ &=0.4x^{2}+0.2x + 0.4 \end{align*}$$

\]

Answer:

$0.4x^{2}+0.2x + 0.4$