Chapter 4.6, Problem 42AYU

To determine

To explain:

The following facts are contradictory:

The polynomial function: $f\left(x\right)$

Degree of polynomial: $3$

Zeroes of polynomial functions: $2,i,3+i$

Answer to Problem 42AYU

As complex zeroes must present in the pair conjugates for the polynomial function $f\left(x\right)$ , the given detailed facts are considered as contradictory.

Explanation of Solution

Given information:

The polynomial function: $f\left(x\right)$

Degree of polynomial: $3$

Zeroes of polynomial functions: $2,i,3+i$

The polynomial function $f\left(x\right)$ has degree of $3$ . The real numbers are the coefficient of the given polynomial. There is three zeroes of the polynomial present as $2,i,3+i$ .

Know that:

The conjugate pair theorem 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:

Here is polynomial of the degree $3$ which has real numbers as its coefficients. From the given three zeroes $i$ and $3+i$ are the complex zeroes.

Now according the theorem if the complex number $i$ is zero of the given polynomial then its conjugate value should also be the zero of the function which is nothing but $-i$ .

Similarly the conjugate value $3+i$ is $3-i$ . But that is not the case in the given problem.

Hence the given facts are contradictory because the complex zeroes must exit in its conjugate pairs for the given polynomial function which has the real numbers as it coefficients.

Hence, given facts are contradictory.