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

Question

___r^3 + ___r^2 + ___r + ____

Answer 1

To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we will distribute each term in the first factor \((r - 1)\) to each term in the second factor \((r^2 - 2r + 3)\).

1. Distribute \(r\): \[ r \cdot (r^2) = r^3 \] \[ r \cdot (-2r) = -2r^2 \] \[ r \cdot (3) = 3r \]

2. Distribute \(-1\): \[ -1 \cdot (r^2) = -r^2 \] \[ -1 \cdot (-2r) = 2r \] \[ -1 \cdot (3) = -3 \]

Now, we combine all the results: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

Next, combine like terms:

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

Putting it all together, we have: \[ r^3 - 3r^2 + 5r - 3 \]

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

So in the blanks, we have:

• The coefficient of \(r^3\) is \(1\),
• The coefficient of \(r^2\) is \(-3\),
• The coefficient of \(r\) is \(5\),
• The constant term is \(-3\).

The answer is: \[ 1,\ -3,\ 5,\ -3 \]