Find a formula for the polynomial $P(x)$ with
- degree 10
- leading coefficient 1
- root of multiplicity 3 at $x=0$
- root of multiplicity 4 at $x=-3$
- root of multiplicity 3 at $x=-11$
\[
P(x)=
\]
Answer
\(\boxed{P(x) = x^{10} + 45x^{9} + 813x^{8} + 7577x^{7} + 39219x^{6} + 113751x^{5} + 173151x^{4} + 107811x^{3}}\)
Step 1 ：Given a polynomial of degree 10, with leading coefficient 1, and roots at x=0, x=-3, and x=-11 with multiplicities 3, 4, and 3 respectively.
Step 2 ：The polynomial can be written in the form: \(P(x) = a*(x - r1)^{m1} * (x - r2)^{m2} * ... * (x - rn)^{mn}\)
Step 3 ：Substituting the given values into the formula, we get the polynomial: \(P(x) = x^{10} + 45x^{9} + 813x^{8} + 7577x^{7} + 39219x^{6} + 113751x^{5} + 173151x^{4} + 107811x^{3}\)
Step 4 ：\(\boxed{P(x) = x^{10} + 45x^{9} + 813x^{8} + 7577x^{7} + 39219x^{6} + 113751x^{5} + 173151x^{4} + 107811x^{3}}\)
