Question

Multiply to simplify the polynomial expression (r−1)(r^2−2r+3)

Answer 1

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).

Let's multiply each term in the first polynomial \((r - 1)\) by each term in the second polynomial \((r^2 - 2r + 3)\):

1. Multiply \(r\) by \(r^2\):
\[
r \cdot r^2 = r^3
\]

2. Multiply \(r\) by \(-2r\):
\[
r \cdot (-2r) = -2r^2
\]

3. Multiply \(r\) by \(3\):
\[
r \cdot 3 = 3r
\]

4. Multiply \(-1\) by \(r^2\):
\[
-1 \cdot r^2 = -r^2
\]

5. Multiply \(-1\) by \(-2r\):
\[
-1 \cdot (-2r) = 2r
\]

6. Multiply \(-1\) by \(3\):
\[
-1 \cdot 3 = -3
\]

Now, we can combine all these results:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]

Next, we combine like terms:
- The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
- The \(r\) terms: \(3r + 2r = 5r\)

Putting it all together, we have:
\[
r^3 - 3r^2 + 5r - 3
\]

Thus, the simplified polynomial expression is:
\[
\boxed{r^3 - 3r^2 + 5r - 3}
\]