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

Question

___r^3 + ___r^2 + ___r + ___
Fill in the blanks

Answer 1

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

Here’s how it's done step by step:

1. Distribute \(r\) from the first factor: \[ r \cdot (r^2 - 2r + 3) = r^3 - 2r^2 + 3r \]

2. Distribute \(-1\) from the first factor: \[ -1 \cdot (r^2 - 2r + 3) = -r^2 + 2r - 3 \]

3. Now, combine all the terms from both distributions: \[ (r^3 - 2r^2 + 3r) + (-r^2 + 2r - 3) \]

4. Combine like terms:

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

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

Now we can fill in the blanks accordingly: \[ r^3 + (-3)r^2 + 5r + (-3) \]

Thus, the filled blanks are: \(1\) + \(-3\) + \(5\) + \(-3\).