To simplify the polynomial expression \((r-2)(r^2-2r+3)\), we will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions:
\[ (r-2)(r^2-2r+3) = r(r^2-2r+3) - 2(r^2-2r+3) \]
Now, we will distribute \(r\) across the polynomial \(r^2 - 2r + 3\):
\[ r(r^2) + r(-2r) + r(3) = r^3 - 2r^2 + 3r \]
Next, we distribute \(-2\) across the same polynomial:
\[ -2(r^2) - 2(-2r) - 2(3) = -2r^2 + 4r - 6 \]
Now, we combine the two results:
\[ r^3 - 2r^2 + 3r - 2r^2 + 4r - 6 \]
Next, we combine like terms:
- The \(r^2\) terms: \(-2r^2 - 2r^2 = -4r^2\)
- The \(r\) terms: \(3r + 4r = 7r\)
Putting it all together, we have:
\[ r^3 - 4r^2 + 7r - 6 \]
Thus, the simplified polynomial expression is:
\[ \boxed{r^3 - 4r^2 + 7r - 6} \]
5 answers