Question

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

Answer 1

To simplify the polynomial expression \((r-2)(r^2-2r+3)\), we will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions:

\[
(r-2)(r^2-2r+3) = r(r^2-2r+3) - 2(r^2-2r+3)
\]

Now, we will distribute \(r\) across the polynomial \(r^2 - 2r + 3\):

\[
r(r^2) + r(-2r) + r(3) = r^3 - 2r^2 + 3r
\]

Next, we distribute \(-2\) across the same polynomial:

\[
-2(r^2) - 2(-2r) - 2(3) = -2r^2 + 4r - 6
\]

Now, we combine the two results:

\[
r^3 - 2r^2 + 3r - 2r^2 + 4r - 6
\]

Next, we combine like terms:

- The \(r^2\) terms: \(-2r^2 - 2r^2 = -4r^2\)
- The \(r\) terms: \(3r + 4r = 7r\)

Putting it all together, we have:

\[
r^3 - 4r^2 + 7r - 6
\]

Thus, the simplified polynomial expression is:

\[
\boxed{r^3 - 4r^2 + 7r - 6}
\]