To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we will apply the distributive property (also known as the FOIL method for binomials).
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Start by distributing \(r\) across the second polynomial:
\[ r \cdot (r^2 - 2r + 3) = r^3 - 2r^2 + 3r \]
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Next, distribute \(-1\) across the second polynomial:
\[ -1 \cdot (r^2 - 2r + 3) = -r^2 + 2r - 3 \]
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Now, combine the two results:
\[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]
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Combine like terms:
- For \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
- For \(r\) terms: \(3r + 2r = 5r\)
- The constant term: \(-3\)
Putting it all together:
\[ r^3 - 3r^2 + 5r - 3 \]
Thus, the simplified result of \((r - 1)(r^2 - 2r + 3)\) is:
\[ \boxed{r^3 - 3r^2 + 5r - 3} \]