Question

write a polynomial that represents the length of the rectangle. the width is x + 0.6 units. the area is 0.9x^3+0.94x^2 + 0.74x + 0.3 square units. the length is units. (use integers or decimals for any numbers in the expression.)

Explanation:

Step1: Recall area formula

The area of a rectangle is $A = lw$, where $A$ is area, $l$ is length and $w$ is width. We need to find $l=\frac{A}{w}$. Given $A = 0.9x^{3}+0.94x^{2}+0.74x + 0.3$ and $w=x + 0.6$.

Step2: Perform polynomial long - division

We divide the polynomial $0.9x^{3}+0.94x^{2}+0.74x + 0.3$ by $x + 0.6$.
First, divide the leading term of the dividend $0.9x^{3}$ by the leading term of the divisor $x$: $\frac{0.9x^{3}}{x}=0.9x^{2}$.
Multiply $x + 0.6$ by $0.9x^{2}$: $0.9x^{2}(x + 0.6)=0.9x^{3}+0.54x^{2}$.
Subtract this from the dividend: $(0.9x^{3}+0.94x^{2}+0.74x + 0.3)-(0.9x^{3}+0.54x^{2})=0.4x^{2}+0.74x + 0.3$.
Next, divide the leading term of the new dividend $0.4x^{2}$ by the leading term of the divisor $x$: $\frac{0.4x^{2}}{x}=0.4x$.
Multiply $x + 0.6$ by $0.4x$: $0.4x(x + 0.6)=0.4x^{2}+0.24x$.
Subtract this from the new dividend: $(0.4x^{2}+0.74x + 0.3)-(0.4x^{2}+0.24x)=0.5x + 0.3$.
Then, divide the leading term of the new dividend $0.5x$ by the leading term of the divisor $x$: $\frac{0.5x}{x}=0.5$.
Multiply $x + 0.6$ by $0.5$: $0.5(x + 0.6)=0.5x+0.3$.
Subtract this from the new dividend: $(0.5x + 0.3)-(0.5x + 0.3)=0$.

Answer:

$0.9x^{2}+0.4x + 0.5$