Chapter 4.6, Problem 41AYU

To determine

To explain:

The following facts are contradictory:

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

Degree of polynomial: $3$

Zeroes of polynomial functions: $4+i,4-i,2+i$

Answer to Problem 41AYU

In the given polynomial function, no real zeroes is present that’s why the given facts are contradictory.

Explanation of Solution

Given information:

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

Degree of polynomial: $3$

Zeroes of polynomial functions: $4+i,4-i,2+i$

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 $4+i,4-i,2+i$ .

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. All the given three zeroes are complex zeroes. Due to degree of the polynomial is odd number, the given facts are contradictory.

The polynomial must contain at least one real zero, but here there is no real zero is listed, all zeroes are the complex zeroes.

Hence the given facts are contradictory because there should be at least one real zero for a polynomial though it has the odd degree of freedom with the coefficients of the real numbers.

Hence, given facts are contradictory.