Given the function $C(n)=(n-5)(-6 n+1)^{3}(n-2)^{8}$,
its roots are
and their multiplicities are
Answer
\(\boxed{Final Answer:}\) The roots of the function \(C(n)=(n-5)(-6 n+1)^{3}(n-2)^{8}\) are \(5\), \(1/6\), and \(2\). Their multiplicities are \(1\), \(3\), and \(8\) respectively. So, the roots and their multiplicities are \((5, 1)\), \((1/6, 3)\), and \((2, 8)\).
Step 1 ：The given function is \(C(n)=(n-5)(-6 n+1)^{3}(n-2)^{8}\).
Step 2 ：The roots of a polynomial are the values of x for which the polynomial equals zero. In this case, the roots of the function are the values of n that make the function equal to zero.
Step 3 ：The multiplicity of a root is the number of times it appears as a solution, which is represented by the exponent of the factor in the polynomial.
Step 4 ：To find the roots and their multiplicities, we need to set each factor of the polynomial equal to zero and solve for n. The multiplicity of each root is the exponent of the corresponding factor.
Step 5 ：Setting \(n-5=0\), we get \(n=5\). So, the root is 5 and its multiplicity is 1.
Step 6 ：Setting \(-6n+1=0\), we get \(n=1/6\). So, the root is 1/6 and its multiplicity is 3.
Step 7 ：Setting \(n-2=0\), we get \(n=2\). So, the root is 2 and its multiplicity is 8.
Step 8 ：\(\boxed{Final Answer:}\) The roots of the function \(C(n)=(n-5)(-6 n+1)^{3}(n-2)^{8}\) are \(5\), \(1/6\), and \(2\). Their multiplicities are \(1\), \(3\), and \(8\) respectively. So, the roots and their multiplicities are \((5, 1)\), \((1/6, 3)\), and \((2, 8)\).
