To determine the height [tex]\( h \)[/tex] of the rectangular prism given its volume and the area of its base, we use the formula for the volume of a prism, which is given by:
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
Given:
- The volume [tex]\( V \)[/tex] of the rectangular prism: [tex]\( V = x^3 - 3x^2 + 5x - 3 \)[/tex]
- The area of the base [tex]\( A \)[/tex]: [tex]\( A = x^2 - 2 \)[/tex]
To find the height [tex]\( h \)[/tex], we solve for [tex]\( h \)[/tex] in the volume formula:
[tex]\[ h = \frac{V}{A} = \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} \][/tex]
We can perform polynomial long division for the given polynomial expression:
[tex]\[ x^3 - 3x^2 + 5x - 3 \div x^2 - 2 \][/tex]
1. Divide the leading term of the numerator [tex]\( x^3 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex] to get [tex]\( x \)[/tex].
2. Multiply [tex]\( x \)[/tex] by the denominator [tex]\( x^2 - 2 \)[/tex] to get [tex]\( x^3 - 2x \)[/tex].
3. Subtract [tex]\( x^3 - 2x \)[/tex] from [tex]\( x^3 - 3x^2 + 5x - 3 \)[/tex]:
[tex]\[
(x^3 - 3x^2 + 5x - 3) - (x^3 - 2x) = -3x^2 + 7x - 3
\][/tex]
4. Divide the leading term of the new numerator [tex]\( -3x^2 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex] to get [tex]\( -3 \)[/tex].
5. Multiply [tex]\( -3 \)[/tex] by the denominator [tex]\( x^2 - 2 \)[/tex] to get [tex]\( -3x^2 + 6 \)[/tex].
6. Subtract [tex]\( -3x^2 + 6 \)[/tex] from [tex]\( -3x^2 + 7x - 3 \)[/tex]:
[tex]\[
(-3x^2 + 7x - 3) - (-3x^2 + 6) = 7x - 9
\][/tex]
Thus, the polynomial long division gives us:
[tex]\[ \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} = x - 3 + \frac{7x - 9}{x^2 - 2} \][/tex]
Therefore, the height of the prism is:
[tex]\[ x - 3 + \frac{7x - 9}{x^2 - 2} \][/tex]
The correct answer is:
[tex]\[ \boxed{x-3+\frac{7 x-9}{x^2-2}} \][/tex]
