9. What is the least degree of a polynomial with roots \( 3+i, 2 i \), and -5 (with multiplicity 2 ) ?

To find the least degree of the polynomial with the given roots, we first note that complex roots come in conjugate pairs when dealing with polynomials with real coefficients. So, since \( 3+i \) is a root, its conjugate \( 3-i \) must also be a root. The roots we have are: - \( 3+i \) - \( 3-i \) - \( 2i \) - \( -2i \) (since \( 2i \) is also a root, its conjugate must be included) - \( -5 \) with multiplicity 2 (which counts as two roots) Counting these, we have a total of: - 1 root from \( 3+i \) - 1 root from \( 3-i \) - 1 root from \( 2i \) - 1 root from \( -2i \) - 2 roots from \( -5 \) (multiplicity 2) Adding these up gives us: 1 + 1 + 1 + 1 + 2 = 6. Thus, the least degree of the polynomial is 6.