Question

How do you find a third degree polynomial given roots 5 and $2i$ ?

Answer 1

Hint: Try to get three factors as a third degree polynomial is required. Two factors can be obtained for two given roots. For the third factor consider the conjugate of the complex root $2i$ i.e. $-2i$. After getting all the factors multiply them to get the required polynomial.

Complete step by step answer:
For a third degree polynomial, we need at least 3 linear factors.
Since we have two roots 5 and $2i$, so we got two factors as $\left( x-5 \right)$ and $\left( x-2i \right)$
For another factor we have to consider the root $2i$.
As we know, $2i$ is a an complex root
So, its conjugate $-2i$ can act as a root for the remaining factor.
Hence, the factor is $\left( x-\left( -2i \right) \right)=\left( x+2i \right)$
Now multiplying all the three factors we have, we can get the polynomial as
$\left( x-5 \right)\left( x-2i \right)\left( x+2i \right)$
Taking two factors at a time
$\begin{align}
  & \Rightarrow \left( x-5 \right)\left( \left( x-2i \right)\left( x+2i \right) \right) \\
 & \Rightarrow \left( x-5 \right)\left( {{x}^{2}}+2xi-2xi-4{{i}^{2}} \right) \\
\end{align}$
As we know, $i=\sqrt{\left( -1 \right)}$
So, ${{i}^{2}}$ can be written as ${{i}^{2}}={{\left( \sqrt{\left( -1 \right)} \right)}^{2}}=-1$
Putting the values of ${{i}^{2}}$ in our equation, we get
$\begin{align}
  & \Rightarrow \left( x-5 \right)\left( {{x}^{2}}-4\left( -1 \right) \right) \\
 & \Rightarrow \left( x-5 \right)\left( {{x}^{2}}-\left( -4 \right) \right) \\
 & \Rightarrow \left( x-5 \right)\left( {{x}^{2}}+4 \right) \\
\end{align}$
Again taking these two factors
$\begin{align}
  & \Rightarrow \left( x-5 \right)\left( {{x}^{2}}+4 \right) \\
 & \Rightarrow {{x}^{3}}+2x-5{{x}^{2}}-20 \\
 & \Rightarrow {{x}^{3}}-5{{x}^{2}}+2x-20 \\
\end{align}$
This is the required polynomial.

Note:
Two factors can be obtained from the two given roots. For the third factor the conjugate of the complex root should be taken into account. Any constant can also be multiplied to get the required polynomial. For example if we multiply the polynomial by a constant ‘5’ then the polynomial we get
$5\left( {{x}^{3}}-5{{x}^{2}}+2x-20 \right)=5{{x}^{3}}-25{{x}^{2}}+10x-100$
It can also be the required polynomial.