If \( g(x) \) is a polynomial with real coefficients and zeros of 2 (multiplicity 2 ), -5 (multiplicity 2 ), \( 7 i \), and \( 3+6 i \), what is the minimum degree of \( g(x) \) ? The minimum degree of \( g(x) \) is \( \square \),
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To determine the minimum degree of \( g(x) \), we need to account for all the zeros and their multiplicities. The polynomial has the following zeros: - 2 with multiplicity 2 contributes 2 to the degree. - -5 with multiplicity 2 contributes 2 to the degree. - \( 7i \), a complex zero, requires its conjugate \( -7i \) to also be a zero, contributing 2 to the degree. - \( 3 + 6i \), another complex zero, requires its conjugate \( 3 - 6i \) to also be a zero, contributing 2 to the degree. Adding these contributions together, we have: \[ 2 (for\: 2) + 2 (for\: -5) + 2 (for\: 7i \: and \: -7i) + 2 (for\: 3 + 6i \: and \: 3 - 6i) = 8 \] Thus, the minimum degree of \( g(x) \) is \( \boxed{8} \).
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