multiply to simplify the polynomial expressions (r-1)(r^2 - 2r + 3) ___r^3+__r^2+___r+___

Answer 1

To simplify the expression \((r-1)(r^2 - 2r + 3)\), we'll distribute \(r-1\) to each term in the polynomial \(r^2 - 2r + 3\).

1. Multiply \(r\) by each term in \(r^2 - 2r + 3\):

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

   So, \(r(r^2 - 2r + 3) = r^3 - 2r^2 + 3r\).

2. Multiply \(-1\) by each term in \(r^2 - 2r + 3\):

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

   So, \(-1(r^2 - 2r + 3) = -r^2 + 2r - 3\).

3. Now, combine the results from steps 1 and 2: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

   Combine like terms:

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

So, combining everything gives us: \[ r^3 - 3r^2 + 5r - 3 \]

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

In the format requested: ___r^3 + __r^2 + ___r + ___ would be:

1. \(1\) for \(r^3\)
2. \(-3\) for \(r^2\)
3. \(5\) for \(r\)
4. \(-3\) (the constant term)

So, the final answer is: 1, -3, 5, -3