Question

Multiply to simplify the polynomial expression left parenthesis r minus 1 right parenthesis left parenthesis r squared minus 2 r plus 3 right parenthesis

Answer 1

To multiply and simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we can use the distributive property, commonly known as the FOIL method for two binomials.

We'll distribute \( (r - 1) \) across each term in \( (r^2 - 2r + 3) \):

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

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

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

Combine like terms:
- The \( r^2 \) terms: \(-2r^2 - r^2 = -3r^2\)
- The \( r \) terms: \(3r + 2r = 5r\)

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

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