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).

Let’s distribute \(r - 1\) to each term in \(r^2 - 2r + 3\):

\[
(r - 1)(r^2 - 2r + 3) = r(r^2 - 2r + 3) - 1(r^2 - 2r + 3)
\]

Now, let's distribute \(r\):

\[
r(r^2) + r(-2r) + r(3) = r^3 - 2r^2 + 3r
\]

Now, distribute \(-1\):

\[
-1(r^2) + -1(-2r) + -1(3) = -r^2 + 2r - 3
\]

Now, combine both results:

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

Combine like terms:

\[
r^3 + (-2r^2 - r^2) + (3r + 2r) - 3 = r^3 - 3r^2 + 5r - 3
\]

Thus, the simplified polynomial expression is:

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

So, filling in the coefficients for \( ()r^3 + ()r^2 + ()r + () \):

- Coefficient of \(r^3\) is 1
- Coefficient of \(r^2\) is -3
- Coefficient of \(r\) is 5
- Constant term is -3

The final answer is:

1r^3 + (-3)r^2 + 5r + (-3)