How do you write a polynomial with zeros i, -3i, 3i and leading coefficient 1?

Question

1709 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-08T08:14:43

The polynomial with zeros $i$ , $-3i$ and $3i$

is $x^3-ix^2+9x-9i$

A polynomial with zeros $a$ , $b$ and $c$ is

$(x-a)(x-b)(x-c)$

Hence a polynomial with zeros $i$ , $-3i$ and $3i$ is

$(x-i)(x-(-3i))(x-3i)$

= $(x-i)(x+3i)(x-3i)$

= $(x-i)(x^2+9)$

= $x^3-ix^2+9x-9i$

Here leading coefficient (of $x^3$ ) is already one.

Hence, the polynomial with zeros $i$ , $-3i$ and $3i$ is $x^3-ix^2+9x-9i$