To determine the length of the rectangle, we divide the area of the rectangle by its width.

Given:

• Area = $x^3 - 2x^2 - 5x + 6$
• Width = $x - 1$

The length is calculated as: $\text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{x^3 - 2x^2 - 5x + 6}{x - 1}$

Step-by-step solution:

We divide $x^3 - 2x^2 - 5x + 6$ by $x - 1$ using polynomial long division:

1. First term: Divide the leading term $x^3$ by $x$, which gives $x^2$. Multiply $x^2 \cdot (x - 1) = x^3 - x^2$.

   Subtract: $(x^3 - 2x^2 - 5x + 6) - (x^3 - x^2) = -x^2 - 5x + 6$

2. Second term: Divide the leading term $-x^2$ by $x$, which gives $-x$. Multiply $-x \cdot (x - 1) = -x^2 + x$.

   Subtract: $(-x^2 - 5x + 6) - (-x^2 + x) = -6x + 6$

3. Third term: Divide the leading term $-6x$ by $x$, which gives $-6$. Multiply $-6 \cdot (x - 1) = -6x + 6$.

   Subtract: $(-6x + 6) - (-6x + 6) = 0$

Final result:

The quotient is: $\text{Length} = x^2 - x - 6$

Factorization (optional check):

We can factorize $x^2 - x - 6$ to confirm: $x^2 - x - 6 = (x - 3)(x + 2)$

Thus, the length of the rectangle is: $x^2 - x - 6 \quad \text{or equivalently} \quad (x - 3)(x + 2)$

Do you need further explanation, or would you like me to clarify any steps?

Related Questions:

1. What is polynomial long division, and how does it work?
2. How do you check the result of a division with polynomial multiplication?
3. What happens when the divisor does not perfectly divide the dividend?
4. How can synthetic division be used instead of long division?
5. How can you factorize a quadratic polynomial?

Tip: Always verify your polynomial division result by multiplying the quotient and divisor. If correct, the product should equal the original polynomial.