To simplify the expression \((r - 1)(r^2 - 2r + 3)\), we can use the distributive property (also known as the FOIL method for binomials). We will multiply \(r - 1\) by each term in the polynomial \(r^2 - 2r + 3\):
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Multiply \(r\) by each term in \(r^2 - 2r + 3\):
- \(r \cdot r^2 = r^3\)
- \(r \cdot (-2r) = -2r^2\)
- \(r \cdot 3 = 3r\)
So, combining these, we have: \[ r^3 - 2r^2 + 3r \]
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Now multiply \(-1\) by each term in \(r^2 - 2r + 3\):
- \(-1 \cdot r^2 = -r^2\)
- \(-1 \cdot (-2r) = 2r\)
- \(-1 \cdot 3 = -3\)
So, combining these, we have: \[ -r^2 + 2r - 3 \]
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Now, combine all the terms from step 1 and step 2: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]
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Combine like terms:
- The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
- The \(r\) terms: \(3r + 2r = 5r\)
So, we have: \[ r^3 - 3r^2 + 5r - 3 \]
The simplified polynomial expression is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]
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