zeros of cubic polynomial is given.

complex zeros are in pair and one real.

complex zeros are \(\displaystyle{2}+\sqrt{{{2}}}{i},{2}-\sqrt{{{2}}}{i}\)

real zeros is 5.

then factors of the polynomial will be

\(\displaystyle{x}-{5},{x}-{\left({2}+\sqrt{{{2}}}{i}\right)},{x}-{\left({2}-\sqrt{{{2}}}{i}\right)}\)

equation will be

\(\displaystyle{\left({x}-{5}\right)}{\left\lbrace{x}-{\left({2}+\sqrt{{{2}}}{i}\right)}\right\rbrace}{\left\lbrace{x}-{\left({2}-\sqrt{{{2}}}{i}\right)}\right\rbrace}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{x}{\left({2}-\sqrt{{{2}}}{i}\right)}-{x}{\left({2}+\sqrt{{2}}{i}\right)}+{\left({2}+\sqrt{{2}}{i}\right)}{\left({2}-\sqrt{{2}}{i}\right)}\right)}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{x}{\left({2}+\sqrt{{2}}{i}+{2}-\sqrt{{2}}{i}\right)}+{2}^{{2}}-{\left(\sqrt{{2}}{i}\right)}^{{2}}\right)}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{4}{x}+{4}+{2}\right)}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{4}{x}+{4}+{2}\right)}={0}\)

\(\displaystyle{x}^{{3}}-{4}{x}^{{2}}+{6}{x}-{5}{x}^{{2}}+{20}{x}-{30}={0}\)

\(\displaystyle{x}^{{3}}-{9}{x}^{{2}}+{26}{x}-{30}={0}\)

polynomial will be

\(\displaystyle{y}={a}{\left({x}^{{3}}-{9}{x}^{{2}}+{26}{x}-{30}\right)}.\ {a}\ne{0}\)

there are infinite number of polynomials.

complex zeros are in pair and one real.

complex zeros are \(\displaystyle{2}+\sqrt{{{2}}}{i},{2}-\sqrt{{{2}}}{i}\)

real zeros is 5.

then factors of the polynomial will be

\(\displaystyle{x}-{5},{x}-{\left({2}+\sqrt{{{2}}}{i}\right)},{x}-{\left({2}-\sqrt{{{2}}}{i}\right)}\)

equation will be

\(\displaystyle{\left({x}-{5}\right)}{\left\lbrace{x}-{\left({2}+\sqrt{{{2}}}{i}\right)}\right\rbrace}{\left\lbrace{x}-{\left({2}-\sqrt{{{2}}}{i}\right)}\right\rbrace}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{x}{\left({2}-\sqrt{{{2}}}{i}\right)}-{x}{\left({2}+\sqrt{{2}}{i}\right)}+{\left({2}+\sqrt{{2}}{i}\right)}{\left({2}-\sqrt{{2}}{i}\right)}\right)}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{x}{\left({2}+\sqrt{{2}}{i}+{2}-\sqrt{{2}}{i}\right)}+{2}^{{2}}-{\left(\sqrt{{2}}{i}\right)}^{{2}}\right)}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{4}{x}+{4}+{2}\right)}={0}\)

\(\displaystyle{\left({x}-{5}\right)}{\left({x}^{{2}}-{4}{x}+{4}+{2}\right)}={0}\)

\(\displaystyle{x}^{{3}}-{4}{x}^{{2}}+{6}{x}-{5}{x}^{{2}}+{20}{x}-{30}={0}\)

\(\displaystyle{x}^{{3}}-{9}{x}^{{2}}+{26}{x}-{30}={0}\)

polynomial will be

\(\displaystyle{y}={a}{\left({x}^{{3}}-{9}{x}^{{2}}+{26}{x}-{30}\right)}.\ {a}\ne{0}\)

there are infinite number of polynomials.