Multiply to simplify the polynomial expression (r−1)(r^2−2r+3) .(2 points)

Question

r^3+r^2+r+

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).

1. Distribute \(r\) to each term in the second polynomial: \[ r \cdot (r^2) = r^3, \] \[ r \cdot (-2r) = -2r^2, \] \[ r \cdot (3) = 3r. \]

2. Next, distribute \(-1\) to each term in the second polynomial: \[ -1 \cdot (r^2) = -r^2, \] \[ -1 \cdot (-2r) = 2r, \] \[ -1 \cdot (3) = -3. \]

3. Now, combine all these results together: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3. \]

4. Combine like terms:

   • For \( r^2 \): \[ -2r^2 - r^2 = -3r^2, \]
   • For \( r \): \[ 3r + 2r = 5r. \]

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

So, the simplified form of \((r - 1)(r^2 - 2r + 3)\) is:

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