To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we can use the distributive property (also known as the FOIL method for binomials). We will 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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Distribute \(-1\): \[ -1 \cdot r^2 = -r^2 \] \[ -1 \cdot (-2r) = 2r \] \[ -1 \cdot 3 = -3 \]
Now, combine all these results together: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]
Next, we can combine like terms:
- The \( -2r^2 \) and \( -r^2 \) combine to give \( -3r^2 \).
- The \( 3r \) and \( 2r \) combine to give \( 5r \).
Thus, the simplified expression is: \[ r^3 - 3r^2 + 5r - 3 \]
So, the final answer is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]
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