Question

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

We'll distribute \( (r - 1) \) to each term in \( (r^2 - 2r + 3) \):

1. Multiply \( r \) by each term in \( r^2 - 2r + 3 \):
\[
r \cdot r^2 = r^3
\]
\[
r \cdot (-2r) = -2r^2
\]
\[
r \cdot 3 = 3r
\]

2. Multiply \( -1 \) by each term in \( r^2 - 2r + 3 \):
\[
-1 \cdot r^2 = -r^2
\]
\[
-1 \cdot (-2r) = 2r
\]
\[
-1 \cdot 3 = -3
\]

Now combine all these results:
\[
r^3 + (-2r^2) + 3r + (-r^2) + 2r + (-3)
\]

This simplifies to:
\[
r^3 + (-2r^2 - r^2) + (3r + 2r) - 3
\]
\[
r^3 - 3r^2 + 5r - 3
\]

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

Filling in the blanks:
__r^3 + __r^2 + __r + __
will be:
**1, -3, 5, -3**

So the completed expression is:
- \( r^3 + (-3)r^2 + 5r - 3 \) or more simply,
- \( 1, -3, 5, -3 \)