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

1 answer

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).

We will 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, we will distribute \(r\):

  1. \(r \cdot r^2 = r^3\)
  2. \(r \cdot (-2r) = -2r^2\)
  3. \(r \cdot 3 = 3r\)

So,

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

Next, we distribute \(-1\):

  1. \(-1 \cdot r^2 = -r^2\)
  2. \(-1 \cdot (-2r) = 2r\)
  3. \(-1 \cdot 3 = -3\)

Thus,

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

Now combine both results:

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

Next, we combine like terms:

  • The \( r^3 \) term remains: \( r^3 \)
  • The \( -2r^2 - r^2 \) terms combine to: \( -3r^2 \)
  • The \( 3r + 2r \) terms combine to: \( 5r \)
  • The constant term is \(-3\).

Thus, the simplified expression is:

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