To multiply and simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we can use the distributive property, commonly known as the FOIL method for two binomials.
We'll distribute \( (r - 1) \) across each term in \( (r^2 - 2r + 3) \):
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Distribute \( r \): \[ r \cdot r^2 = r^3 \] \[ r \cdot (-2r) = -2r^2 \] \[ r \cdot 3 = 3r \]
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Now distribute \(-1\): \[ -1 \cdot r^2 = -r^2 \] \[ -1 \cdot (-2r) = 2r \] \[ -1 \cdot 3 = -3 \]
Now we combine all these results: \[ r^3 + (-2r^2) + 3r + (-r^2) + 2r - 3 \]
Combine like terms:
- The \( r^2 \) terms: \(-2r^2 - r^2 = -3r^2\)
- The \( r \) terms: \(3r + 2r = 5r\)
Putting it all together: \[ r^3 - 3r^2 + 5r - 3 \]
So, the simplified expression is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]