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Multiply the polynomial expression

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

1 answer

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

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

    So, the result of this part is: \[ r^3 - 2r^2 + 3r \]

  2. Now, distribute \(-1\) to each term in \(r^2 - 2r + 3\): \[ -1 \cdot r^2 = -r^2 \] \[ -1 \cdot (-2r) = 2r \] \[ -1 \cdot 3 = -3 \]

    So, the result of this part is: \[ -r^2 + 2r - 3 \]

  3. Now combine the results from both steps: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

  4. Combine like terms:

    • \(r^3\) is alone.
    • Combine the \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\).
    • Combine the \(r\) terms: \(3r + 2r = 5r\).
    • Constant term: \(-3\).

Putting it all together, the final expression is: \[ r^3 - 3r^2 + 5r - 3 \]

So, the result of multiplying \((r - 1)(r^2 - 2r + 3)\) is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]