Chapter 4.6, Problem 43AYU

To determine

To explain:

The polynomial function $f\left(x\right)$ must contain real number as its remaining zeroes.

Answer to Problem 43AYU

The fourth zero value of the given polynomial must be a complex conjugate pair and real number.

Explanation of Solution

The polynomial function $f\left(x\right)$ has degree of $4$ . The real numbers are the coefficient of the given polynomial. There are four zeroes of the polynomial present, three of them are $2,1+2i$ and $1-2i$ .

Now, need to find the missing fourth zeroes of the polynomial function and explained it is real number.

Here, need to use the conjugate pair theorem which states that if a polynomial function $f\left(x\right)$ has the real numbers as its coefficient and $r=a+ib$ is z zero of the given $f$ , then the value of the complex conjugate is also zero of given $f$ .

The complex conjugate is represented by $\overline{r}=a-ib$ . Now, apply this theorem to given statement:

Consider the fourth zero value of the function $f$ is a complex number $a+ib$ . According to theorem, its conjugate value should also be the zero of the function which is nothing but $a-ib$ .

Therefore the fourth zero value of the given polynomial must be a complex conjugate pair and real number.

Hence, given polynomial has the fourth zero value as real number.