Question

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

Answer 1

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

Let's expand it step-by-step:

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

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

Now, combine all the terms together:

\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]

Now, 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: \(-3\)

Putting it all together, we get:

\[
r^3 - 3r^2 + 5r - 3
\]

Now, filling in the boxes as requested:

\[
\boxed{1}r^3 + \boxed{-3}r^2 + \boxed{5}r + \boxed{-3}
\]

Thus, the simplified polynomial expression is:

\[
r^3 - 3r^2 + 5r - 3
\]