Chapter 4.5, Problem 8AYU

To determine

The following sentence needs to decide as true or false:

There are exactly three real zeroes available for every polynomial function whose degree of real coefficient is $3$ .

Answer to Problem 8AYU

False. The correct statement is there are exactly $3$ complex zeroes for the polynomial who has the degree of $3$ with real coefficients.

Explanation of Solution

Given:

Degree of function: $3$

Know that:

The polynomial function $f\left(x\right)$ has degree of $3$ . The real numbers are the coefficient of the given polynomial.

The problem states that it must include real numbers as its zeroes but it is not possible because the zeroes of the given function are three in number, there is chance; zeroes of the function is combination of two complex and one real, or one complex or two real zeroes.

Because as per the Fundamental theorem of the Algebra:

There are exactly $n$ complex zeroes for the polynomial who has the degree of $n$ with real coefficients.

Conclusion:

Hence, the given statement is false.