Question

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

Answer 1

To simplify the expression \((r - 1)(r^2 - 2r + 3)\), we can use the distributive property (also known as the FOIL method when dealing with binomials) to multiply each term in the first polynomial by each term in the second polynomial.

Let's compute it step by step:

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

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

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

4. Combine like terms:
- The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
- The \(r\) terms: \(3r + 2r = 5r\)
- The constant term remains \(-3\)

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

To fill in the blanks:
\[
\_r^3 + \_r^2 + \_r + \_
\]

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 \(-3\).

So the answer is:
\[
1r^3 - 3r^2 + 5r - 3
\]

Thus, the blanks can be filled as:
1, -3, 5, -3.