Question

Multiply to simplify the polynomial expression (r-1) (r²-2r+3)

Answer 1

To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we will use the distributive property (also known as the FOIL method for binomials).

We will distribute \( (r - 1) \) to each term in \( (r^2 - 2r + 3) \):

\[
(r - 1)(r^2 - 2r + 3) = r(r^2 - 2r + 3) - 1(r^2 - 2r + 3)
\]

Now, we will distribute \(r\):

1. \(r \cdot r^2 = r^3\)
2. \(r \cdot (-2r) = -2r^2\)
3. \(r \cdot 3 = 3r\)

So,

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

Next, we distribute \(-1\):

1. \(-1 \cdot r^2 = -r^2\)
2. \(-1 \cdot (-2r) = 2r\)
3. \(-1 \cdot 3 = -3\)

Thus,

\[
-1(r^2 - 2r + 3) = -r^2 + 2r - 3
\]

Now combine both results:

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

Next, we combine like terms:

- The \( r^3 \) term remains: \( r^3 \)
- The \( -2r^2 - r^2 \) terms combine to: \( -3r^2 \)
- The \( 3r + 2r \) terms combine to: \( 5r \)
- The constant term is \(-3\).

Thus, the simplified expression is:

\[
\boxed{r^3 - 3r^2 + 5r - 3}
\]