To determine the height of the rectangular prism given its volume and the area of its base, we start with the relationship that the volume of a rectangular prism is the product of its base area and height.
Let's denote:
- [tex]\( V \)[/tex] as the volume of the rectangular prism, which is [tex]\( x^3 - 3x^2 + 5x - 3 \)[/tex].
- [tex]\( A \)[/tex] as the area of the base of the prism, which is [tex]\( x^2 - 2 \)[/tex].
The relationship between the volume [tex]\( V \)[/tex], base area [tex]\( A \)[/tex], and height [tex]\( h \)[/tex] is given by the formula:
[tex]\[ V = A \times h \][/tex]
We need to find the height [tex]\( h \)[/tex]. To do this, we solve for [tex]\( h \)[/tex] by dividing the volume [tex]\( V \)[/tex] by the base area [tex]\( A \)[/tex]:
[tex]\[ h = \frac{V}{A} = \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} \][/tex]
By performing the division, we get:
[tex]\[ h = \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} \][/tex]
This fraction can be simplified by polynomial division. Let's break down the polynomial division step by step:
1. Divide the leading term of the numerator [tex]\( x^3 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex]:
[tex]\[\frac{x^3}{x^2} = x\][/tex]
2. Multiply [tex]\( x \)[/tex] by the entire denominator [tex]\( x^2 - 2 \)[/tex]:
[tex]\[ x \cdot (x^2 - 2) = x^3 - 2x \][/tex]
3. Subtract this product from the original numerator:
[tex]\[
(x^3 - 3x^2 + 5x - 3) - (x^3 - 2x) = -3x^2 + 7x - 3
\][/tex]
4. Divide the new leading term [tex]\( -3x^2 \)[/tex] by the leading term [tex]\( x^2 \)[/tex]:
[tex]\[\frac{-3x^2}{x^2} = -3\][/tex]
5. Multiply [tex]\(-3\)[/tex] by the denominator [tex]\( x^2 - 2 \)[/tex]:
[tex]\[ -3 \cdot (x^2 - 2) = -3x^2 + 6 \][/tex]
6. Subtract this product from the previous result:
[tex]\[
(-3x^2 + 7x - 3) - (-3x^2 + 6) = 7x - 9
\][/tex]
So, after the polynomial division, the height is expressed as:
[tex]\[ h = x - 3 + \frac{7x - 9}{x^2 - 2} \][/tex]
Thus, the height of the prism is:
[tex]\[ x - 3 + \frac{7x - 9}{x^2 - 2} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{x - 3 + \frac{7 x - 9}{x^2 - 2}} \][/tex]
