Question

1. the area, a(x), of a rectangle is represented by the polynomial 2x² - x - 6. a) if the height of the rectangle is x - 2, what is the width in terms of x? b) if the height of the rectangle were changed to x - 3, what would the remainder of the quotient be? what does this remainder represent?

Explanation:

Step1: Recall area formula for rectangle

The area of a rectangle is $A = \text{height}\times\text{width}$. Given $A(x)=2x^{2}-x - 6$ and height $h(x)=x - 2$, we find width $w(x)$ by $w(x)=\frac{A(x)}{h(x)}$.

Step2: Factor the area polynomial

Factor $2x^{2}-x - 6$. We have $2x^{2}-x - 6=2x^{2}-4x + 3x-6=2x(x - 2)+3(x - 2)=(2x + 3)(x - 2)$.

Step3: Find the width for part a

Since $w(x)=\frac{A(x)}{h(x)}$ and $A(x)=(2x + 3)(x - 2)$, $h(x)=x - 2$, then $w(x)=\frac{(2x + 3)(x - 2)}{x - 2}=2x+3$ (for $x
eq2$).

Step4: Use polynomial long - division for part b

Divide $2x^{2}-x - 6$ by $x - 3$ using polynomial long - division.
\[

$$\begin{align*} 2x^{2}-x - 6&=2x(x - 3)+6x - x-6\\ &=2x(x - 3)+5x - 6\\ &=2x(x - 3)+5(x - 3)+15 - 6\\ &=2x(x - 3)+5(x - 3)+9 \end{align*}$$

\]
So, $\frac{2x^{2}-x - 6}{x - 3}=2x + 5+\frac{9}{x - 3}$. The remainder is 9. In the context of the rectangle, if we consider the division of the area polynomial by the new height polynomial, the remainder represents the non - divisible part of the area with respect to the new height.

Answer:

a) $2x + 3$
b) Remainder: 9. It represents the non - divisible part of the area with respect to the new height.