Find the polynomial function $f (x)$ of least degree having only real coefficients and zeros as given.
Assume multiplicity $1$ .
$0, - i and 2 + i$ .

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Solution

Find the polynomial function $f (x)$ of least degree with real coefficients and and given zeros:
$\begin{array}{rcl} If x & = & a, b, c . . . arethezerosofafunction f (x) \\ \Rightarrow & f (x) = (x - a) (x - b) (x - c) . . . . . \end{array}$
The given zeros are $0, - i and 2 + i$ .
$∵$ The complex roots are exists in conjugate pair. So the zeros are $0, - i, i and 2 + i, 2 - i$ .
$\Rightarrow f (x) = (x - 0) [(x + i) (x - i)] [(x - 2 - i) (x - 2 + i)]$
Simplify the equation $f (x) = (x - 0) [(x + i) (x - i)] [(x - 2 - i) (x - 2 + i)]$ :
$\begin{array}{rcl} f (x) & = & x [(x - i) (x + i)] [(x - 2 - i) (x - 2 + i)] \\ = & x (x^{2} - i^{2}) [{(x - 2)}^{2} - i^{2}] \\ = & x (x^{2} + 1) (x^{2} - 4 x + 5) \\ = & x (x^{4} - 4 x^{3} + 5 x^{2} + x^{2} - 4 x + 5) \\ = & x (x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 5) \\ = & x^{5} - 4 x^{4} + 6 x^{3} - 4 x^{2} + 5 x \end{array}$
Hence, the required polynomial function is $f (x) = x^{5} - 4 x^{4} + 6 x^{3} - 4 x^{2} + 5 x$ .

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