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

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To determine the least degree of a polynomial with the given roots, we'll assume the polynomial has real coefficients. This assumption is important because if the coefficients are real, then any complex roots must come in conjugate pairs.

**Given Roots:**
- $$ 3 + i $$
- $$ 2i $$
- $$ -5 $$ (with multiplicity 2)

**Including Conjugate Pairs:**
1. The root $$ 3 + i $$ implies the conjugate root $$ 3 - i $$.
2. The root $$ 2i $$ implies the conjugate root $$ -2i $$.

**Complete List of Roots:**
- $$ 3 + i $$
- $$ 3 - i $$
- $$ 2i $$
- $$ -2i $$
- $$ -5 $$ (with multiplicity 2)

**Counting the Roots:**
- $$ 3 + i $$ and $$ 3 - i $$: 2 roots
- $$ 2i $$ and $$ -2i $$: 2 roots
- $$ -5 $$ with multiplicity 2: 2 roots

**Total:** $$ 2 + 2 + 2 = 6 $$ roots

Therefore, the least degree of the polynomial is **6**.

**Answer:** 6
The least degree of the polynomial is 6.

What is the least degree of a polynomial with roots , and -5 (with multiplicity 2 ) ?

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