Sure, let's solve this step-by-step.
Given:
1. The area of the rectangle door can be described by the trinomial [tex]\(A = x^2 - 14x\)[/tex].
2. The width of the door is given by [tex]\(W = x - 9\)[/tex].
We need to find the length [tex]\(L\)[/tex] of the door.
### Step-by-Step Solution:
1. Area of a Rectangle Formula:
The area [tex]\(A\)[/tex] of a rectangle is calculated using the formula:
[tex]\[
A = \text{length} \times \text{width}
\][/tex]
We are provided with:
[tex]\[
A = x^2 - 14x
\][/tex]
[tex]\[
W = x - 9
\][/tex]
2. Express Length in Terms of Area and Width:
From the formula for the area of a rectangle, we can rearrange it to solve for the length [tex]\(L\)[/tex]:
[tex]\[
L = \frac{A}{W}
\][/tex]
3. Substitute the Given Formulas:
We substitute in the given expressions for [tex]\(A\)[/tex] and [tex]\(W\)[/tex]:
[tex]\[
L = \frac{x^2 - 14x}{x - 9}
\][/tex]
4. Polynomial Division:
To simplify [tex]\(\frac{x^2 - 14x}{x - 9}\)[/tex], we perform polynomial division.
- Divide the leading term of the numerator [tex]\(x^2\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex]:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
- Multiply [tex]\(x\)[/tex] by the denominator [tex]\(x - 9\)[/tex]:
[tex]\[
x(x - 9) = x^2 - 9x
\][/tex]
- Subtract [tex]\(x^2 - 9x\)[/tex] from [tex]\(x^2 - 14x\)[/tex]:
[tex]\[
(x^2 - 14x) - (x^2 - 9x) = -14x + 9x = -5x
\][/tex]
- The result is [tex]\(-5x\)[/tex]:
[tex]\[
\frac{-5x}{x - 9}
\][/tex]
- Divide [tex]\(-5x\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex]:
[tex]\[
-5
\][/tex]
- Multiply [tex]\(-5\)[/tex] by the denominator [tex]\(x - 9\)[/tex]:
[tex]\[
-5(x - 9) = -5x + 45
\][/tex]
- Subtract [tex]\(-5x + 45\)[/tex] from [tex]\(-5x\)[/tex]:
[tex]\[
-5x - (-5x + 45) = 0 - 45 = -45
\][/tex]
Combining all parts, we get:
[tex]\[
x - 5 + 0
\][/tex]
The polynomial division shows that [tex]\(x^2 - 14x\)[/tex] divided by [tex]\(x - 9\)[/tex] simplifies nicely to [tex]\(x - 5\)[/tex].
### Conclusion:
Therefore, the length of the door [tex]\(L\)[/tex] is:
[tex]\[
\boxed{x - 5}
\][/tex]
