Question

write a polynomial that represents the length of the rectangle. the width is x + 0.9 units. the area is 0.8x³ + 1.22x² + 0.95x + 0.45 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: Perform polynomial long - division

We divide the polynomial $0.8x^{3}+1.22x^{2}+0.95x + 0.45$ by the polynomial $x + 0.9$.
\[

$$\begin{align*} 0.8x^{3}+1.22x^{2}+0.95x + 0.45&=0.8x^{3}+0.72x^{2}+0.5x^{2}+0.45x+0.5x + 0.45\\ &=0.8x^{2}(x + 0.9)+0.5x(x + 0.9)+0.5(x + 0.9)\\ &=(x + 0.9)(0.8x^{2}+0.5x + 0.5) \end{align*}$$

\]
So, $\frac{0.8x^{3}+1.22x^{2}+0.95x + 0.45}{x + 0.9}=0.8x^{2}+0.5x + 0.5$

Answer:

$0.8x^{2}+0.5x + 0.5$