Multiply to simplify the polynomial expression (r−1)(r^2−2r+3)

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 multiply each term in the first polynomial \((r - 1)\) by each term in the second polynomial \((r^2 - 2r + 3)\):

1. Multiply \(r\) by \(r^2\): \[ r \cdot r^2 = r^3 \]

2. Multiply \(r\) by \(-2r\): \[ r \cdot (-2r) = -2r^2 \]

3. Multiply \(r\) by \(3\): \[ r \cdot 3 = 3r \]

4. Multiply \(-1\) by \(r^2\): \[ -1 \cdot r^2 = -r^2 \]

5. Multiply \(-1\) by \(-2r\): \[ -1 \cdot (-2r) = 2r \]

6. Multiply \(-1\) by \(3\): \[ -1 \cdot 3 = -3 \]

Now, we can combine all these results: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

Next, we combine like terms:

• The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
• The \(r\) terms: \(3r + 2r = 5r\)

Putting it all together, we have: \[ r^3 - 3r^2 + 5r - 3 \]

Thus, the simplified polynomial expression is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]