Chapter 4.6, Problem 44AYU

To determine

To explain:

The polynomial function $f\left(x\right)$ must contain one real number as zeroes for the function and the value of the missing zeroes.

Answer to Problem 44AYU

One of the missing value of zeroes of the given polynomial is $4+i$ .

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 three zeroes of the polynomial present, two of them are $-3$ and $4-i$ .

Now, need to find the missing one 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:

If $4-i$ is zero of the given polynomial function $f$ , then its complex conjugate $4+i$ should also be zero of the function.

Therefore,

The three zeroes of the polynomial function $f$ are listed as:

  $-3,4-i,4+i$

Consider the fourth zero value of the function $f$ is a complex number $a+ib$ . According to theorem, here one real number is zero for given function is which is $-3$ , so its conjugate should also be the zero of the function which is nothing but $a-ib$ .

But it is not possible because one of the zero values given for function $-3$ is a real number. Therefore the fourth zero value of the given polynomial must be a complex conjugate pair and real number. The missing zero value is $4+i$ .