The area of a rectangle is given by the formula:

$\text{Area} = \text{Length} \times \text{Width}$

We are given:

$\text{Area} = x^2 - 21x + 108$ $\text{Width} = x - 9$

To find the length, we need to divide the area by the width:

$\text{Length} = \frac{x^2 - 21x + 108}{x - 9}$

Step 1: Factor the Numerator

We factor $x^2 - 21x + 108$. We need two numbers that multiply to 108 and add up to -21. The numbers -12 and -9 satisfy this:

$x^2 - 21x + 108 = (x - 9)(x - 12)$

Step 2: Simplify the Expression

$\frac{(x - 9)(x - 12)}{x - 9}$

Cancel out $x - 9$:

$x - 12$

Final Answer:

$\text{Length} = x - 12$

Would you like a more detailed explanation?

Related Questions:

1. How do you factor quadratic expressions like $x^2 - 21x + 108$?
2. What happens if $x = 9$ in this problem?
3. How do you perform polynomial division for cases where factoring is difficult?
4. What is the perimeter of the rectangle in terms of $x$?
5. If $x = 15$, what is the actual length and width of the rectangle?

Tip:

When factoring a quadratic expression $ax^2 + bx + c$, look for two numbers that multiply to $ac$ and sum to $b$.