How do you write a polynomial function in standard form with real coefficients whose zeros include 3, 5i, and -5i?

May 7, 2016

The polynomial function in standard form with real coefficients whose zeros include $3$, $5 i$, and $- 5 i$ is ${x}^{3} - 3 {x}^{2} + 25 x - 75$

Explanation:

A polynomial function with zeros as $a$, $b$ and $c$ would be

$\left(x - a\right) \left(x - b\right) \left(x - c\right)$

Hence a polynomial function with zeros as $3$, $5 i$ and $- 5 i$ would be

$\left(x - 3\right) \left(x - 5 i\right) \left(x - \left(- 5 i\right)\right)$ or

$\left(x - 3\right) \left(x - 5 i\right) \left(x + 5 i\right)$ or

$\left(x - 3\right) \left({x}^{2} - {\left(5 i\right)}^{2}\right)$ or

$\left(x - 3\right) \left({x}^{2} - 25 {i}^{2}\right)$ or

$\left(x - 3\right) \left({x}^{2} + 25\right)$ or

${x}^{3} - 3 {x}^{2} + 25 x - 75$