Question
Multiply to simplify the polynomial expression (r−1)(r2−2r+3)
(
𝑟
−
1
)
(
𝑟
2
−
2
𝑟
+
3
)
.(2 points)
r3+
𝑟
3
+
r2+
𝑟
2
+
r+
𝑟
+
(
𝑟
−
1
)
(
𝑟
2
−
2
𝑟
+
3
)
.(2 points)
r3+
𝑟
3
+
r2+
𝑟
2
+
r+
𝑟
+
Answers
GPT-4o mini
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).
1. Start by distributing \(r\) across the second polynomial:
\[
r \cdot (r^2 - 2r + 3) = r^3 - 2r^2 + 3r
\]
2. Next, distribute \(-1\) across the second polynomial:
\[
-1 \cdot (r^2 - 2r + 3) = -r^2 + 2r - 3
\]
3. Now, combine the two results:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]
4. 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:
\[
\boxed{r^3 - 3r^2 + 5r - 3}
\]
1. Start by distributing \(r\) across the second polynomial:
\[
r \cdot (r^2 - 2r + 3) = r^3 - 2r^2 + 3r
\]
2. Next, distribute \(-1\) across the second polynomial:
\[
-1 \cdot (r^2 - 2r + 3) = -r^2 + 2r - 3
\]
3. Now, combine the two results:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]
4. 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:
\[
\boxed{r^3 - 3r^2 + 5r - 3}
\]