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

Question

.(2 points)
___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).

1. Distribute \(r\) in \((r - 1)\):

   • \(r \cdot r^2 = r^3\)
   • \(r \cdot (-2r) = -2r^2\)
   • \(r \cdot 3 = 3r\)

   So we get: \[ r(r^2 - 2r + 3) = r^3 - 2r^2 + 3r \]

2. Distribute \(-1\) in \((r - 1)\):

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

   So we get: \[ -1(r^2 - 2r + 3) = -r^2 + 2r - 3 \]

3. Now combine all the terms from both distributions: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

4. Combine like terms:

   • \(r^3\)
   • \(-2r^2 - r^2 = -3r^2\)
   • \(3r + 2r = 5r\)
   • And \(-3\)

Thus, the simplified polynomial is: \[ r^3 - 3r^2 + 5r - 3 \]

So, in the format requested: ___r^3 + ___r^2 + ___r + ___

You would fill it in as: 1 \(r^3\), -3 \(r^2\), 5 \(r\), -3.