To determine the length of the rectangle given the area and the width, we proceed as follows:
Given:
- Area of the rectangle: \( A = x^3 - 5x^2 + 3x - 15 \)
- Width of the rectangle: \( W = x^2 + 3 \)
- Area \( = \) Length \( \times \) Width
We need to find the Length (\( L \)) of the rectangle. The formula can be rearranged to:
[tex]\[ L = \frac{A}{W} \][/tex]
We substitute the given expressions for \( A \) and \( W \):
[tex]\[ L = \frac{x^3 - 5x^2 + 3x - 15}{x^2 + 3} \][/tex]
To simplify this division, we perform polynomial division. Here are the detailed steps:
1. Divide the leading term of the numerator \( (x^3) \) by the leading term of the denominator \( (x^2) \):
[tex]\[ \frac{x^3}{x^2} = x \][/tex]
2. Multiply \( x \) by \( x^2 + 3 \) and subtract the result from the original polynomial:
[tex]\[ (x)(x^2 + 3) = x^3 + 3x \][/tex]
[tex]\[ (x^3 - 5x^2 + 3x - 15) - (x^3 + 3x) = x^3 - 5x^2 + 3x - 15 - x^3 - 3x \][/tex]
[tex]\[ = -5x^2 - 6x - 15 \][/tex]
3. Divide the new leading term of the result \( (-5x^2) \) by the leading term of the denominator \( (x^2) \):
[tex]\[ \frac{-5x^2}{x^2} = -5 \][/tex]
4. Multiply \( -5 \) by \( x^2 + 3 \) and subtract the result from the polynomial obtained in the previous step:
[tex]\[ (-5)(x^2 + 3) = -5x^2 - 15 \][/tex]
[tex]\[ (-5x^2 - 6x - 15) - (-5x^2 - 15) = -5x^2 - 6x - 15 + 5x^2 + 15 \][/tex]
[tex]\[ = -6x \][/tex]
The result is that the remainder is \(-6x\) and it's much smaller degree than the divisor. This confirms that the correct quotient is:
[tex]\[ L = x - 5 \][/tex]
Thus, the length of the rectangle is:
[tex]\[ \boxed{x - 5} \][/tex]
