The volume of a rectangular prism is x cubed minus 3x squared plus 5x minus 3, and the area of its base is x squared minus 2. Since volume is the product of base area and height, what is the height of the prism?

Step-by-step solution

1. 1

   Turn the geometric relation into a division. From $V=A\cdot h$,

   $h=\frac{V}{A}=\frac{x^3-3x^2+5x-3}{x^2-2}$

   The divisor is a quadratic, so synthetic division does not apply — this needs full polynomial long division.

2. 2

   First division step: match the leading terms. $x^3\div x^2=x$, so the first quotient term is $x$. Multiply and subtract:

   $x(x^2-2)=x^3-2x$
   $(x^3-3x^2+5x-3)-(x^3-2x)=-3x^2+7x-3$

3. 3

   Second division step. $-3x^2\div x^2=-3$, so the next quotient term is $-3$. Multiply and subtract:

   $-3(x^2-2)=-3x^2+6$
   $(-3x^2+7x-3)-(-3x^2+6)=7x-9$

   The remainder $7x-9$ has degree $1$, below the divisor's degree $2$, so the division stops here.

4. 4

   Write the height as quotient plus remainder over divisor.

   $h=x-3+\frac{7x-9}{x^2-2}$

   Because the remainder is not the zero polynomial, the height is a rational expression, not a polynomial — the numbers in this problem simply do not divide evenly.

5. 5

   Verify by multiplying back. Reassembling $A\cdot h$ should return $V$:

   $(x^2-2)(x-3)+(7x-9)=x^3-3x^2-2x+6+7x-9=x^3-3x^2+5x-3\ \checkmark$

   At $x=5$ both sides give $72$, confirming the identity numerically as well.

6. 6

   Spot-check the height at a convenient value. At $x=2$: $V=8-12+10-3=3$ and $A=4-2=2$, so the true height is $\tfrac32$. The formula gives

   $(2-3)+\frac{14-9}{4-2}=-1+2.5=1.5\ \checkmark$