To find the width of the rectangular field, we need to divide the polynomial that represents the area of the field by the polynomial that represents the length of the field.
The area \( A \) of the rectangular field is given by: \[ A = x^3 + x^2 - 14x - 24 \] The length \( L \) is given by: \[ L = x - 4 \] We need to find the width \( W \) such that: \[ W = \frac{A}{L} \]
To perform the polynomial long division of \( A \) by \( L \):
-
Divide the leading term of \( A \) (which is \( x^3 \)) by the leading term of \( L \) (which is \( x \)): \[ \frac{x^3}{x} = x^2 \]
-
Multiply \( L \) by \( x^2 \): \[ (x - 4) \cdot x^2 = x^3 - 4x^2 \]
-
Subtract this result from \( A \): \[ A - (x^3 - 4x^2) = (x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = 5x^2 - 14x - 24 \]
-
Now, take the new polynomial \( 5x^2 - 14x - 24 \) and divide the leading term \( 5x^2 \) by the leading term of \( L \): \[ \frac{5x^2}{x} = 5x \]
-
Multiply \( L \) by \( 5x \): \[ (x - 4) \cdot 5x = 5x^2 - 20x \]
-
Subtract this from \( 5x^2 - 14x - 24 \): \[ (5x^2 - 14x - 24) - (5x^2 - 20x) = 6x - 24 \]
-
Next, take \( 6x - 24 \) and divide the leading term \( 6x \) by the leading term of \( L \): \[ \frac{6x}{x} = 6 \]
-
Multiply \( L \) by \( 6 \): \[ (x - 4) \cdot 6 = 6x - 24 \]
-
Subtract this from \( 6x - 24 \): \[ (6x - 24) - (6x - 24) = 0 \]
Thus, the division is complete. The quotient of the division gives us the width of the field: \[ W = x^2 + 5x + 6 \]
We can also factor this expression further: \[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
Therefore, the expression that represents the width of the field is: \[ W = x^2 + 5x + 6 \] or, factored, \[ W = (x + 2)(x + 3) \]