Question

Multiply to simplify the polynomial expression (r−1)(r2−2r+3)

__ r ^3 + __ r^2 + __r + __

Answer 1

To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we can use the distributive property (also known as the FOIL method for binomials).

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

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

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

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

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

Putting it all together, the simplified polynomial expression is:
\[
r^3 - 3r^2 + 5r - 3
\]

So the blanks in the expression \( r^3 + \_ r^2 + \_ r + \_ \) can be filled as follows:
- \(1\) for \(r^3\)
- \(-3\) for \(r^2\)
- \(5\) for \(r\)
- \(-3\) for the constant term

Thus, the final answer is:
\[
1 \quad -3 \quad 5 \quad -3
\]