To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we will use the distributive property (also known as the FOIL method for binomials).
We will distribute \( (r - 1) \) to each term in \( (r^2 - 2r + 3) \):
\[ (r - 1)(r^2 - 2r + 3) = r(r^2 - 2r + 3) - 1(r^2 - 2r + 3) \]
Now, we will distribute \(r\):
- \(r \cdot r^2 = r^3\)
- \(r \cdot (-2r) = -2r^2\)
- \(r \cdot 3 = 3r\)
So,
\[ r(r^2 - 2r + 3) = r^3 - 2r^2 + 3r \]
Next, we distribute \(-1\):
- \(-1 \cdot r^2 = -r^2\)
- \(-1 \cdot (-2r) = 2r\)
- \(-1 \cdot 3 = -3\)
Thus,
\[ -1(r^2 - 2r + 3) = -r^2 + 2r - 3 \]
Now combine both results:
\[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]
Next, we combine like terms:
- The \( r^3 \) term remains: \( r^3 \)
- The \( -2r^2 - r^2 \) terms combine to: \( -3r^2 \)
- The \( 3r + 2r \) terms combine to: \( 5r \)
- The constant term is \(-3\).
Thus, the simplified expression is:
\[ \boxed{r^3 - 3r^2 + 5r - 3} \]