Question

multiply to simplify the polynomial expression then fill in the blanks (_)
(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 can use the distributive property (also known as the FOIL method for binomials).

1. Multiply \(r\) by each term in \(r^2 - 2r + 3\):
- \(r \cdot r^2 = r^3\)
- \(r \cdot (-2r) = -2r^2\)
- \(r \cdot 3 = 3r\)

2. Multiply \(-1\) by each term in \(r^2 - 2r + 3\):
- \(-1 \cdot r^2 = -r^2\)
- \(-1 \cdot (-2r) = 2r\)
- \(-1 \cdot 3 = -3\)

Now, combine all these results:
- The \(r^3\) term: \(r^3\)
- The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
- The \(r\) terms: \(3r + 2r = 5r\)
- The constant term: \(-3\)

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

Now we can fill in the blanks:
\[
\_r^3 + \_-3r^2 + \_5r + \_-3
\]

So the final result is:
\[
\boxed{1}, \boxed{-3}, \boxed{5}, \boxed{-3}
\]