Question

Multiply to simplify the polynomial expression (r - 1) (r^2 - 2r + 3).
___r^3 + ___r^2 + ___r + ____

Answer 1

To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we will distribute each term in the first factor \((r - 1)\) to each term in the second factor \((r^2 - 2r + 3)\).

1. Distribute \(r\):
\[
r \cdot (r^2) = r^3
\]
\[
r \cdot (-2r) = -2r^2
\]
\[
r \cdot (3) = 3r
\]

2. Distribute \(-1\):
\[
-1 \cdot (r^2) = -r^2
\]
\[
-1 \cdot (-2r) = 2r
\]
\[
-1 \cdot (3) = -3
\]

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

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

So in the blanks, we have:
- The coefficient of \(r^3\) is \(1\),
- The coefficient of \(r^2\) is \(-3\),
- The coefficient of \(r\) is \(5\),
- The constant term is \(-3\).

The answer is:
\[
1,\ -3,\ 5,\ -3
\]