Question

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

___r^3 + ___r^2 + ___r + ___
Fill in the blanks

Answer 1

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

Here’s how it's done step by step:

1. Distribute \(r\) from the first factor:
\[
r \cdot (r^2 - 2r + 3) = r^3 - 2r^2 + 3r
\]

2. Distribute \(-1\) from the first factor:
\[
-1 \cdot (r^2 - 2r + 3) = -r^2 + 2r - 3
\]

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

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

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

Now we can fill in the blanks accordingly:
\[
r^3 + (-3)r^2 + 5r + (-3)
\]

Thus, the filled blanks are:
___\(1\)___ + ___\(-3\)___ + ___\(5\)___ + ___\(-3\)___.