Chapter 5.6, Problem 45PPSE

Solution

To state : The required polynomial function as per the given information.

The required polynomial function is $y=x^4+3x^2+2$ .

Given information :

The polynomial function has four possible rational zeros but no actual rational zeros.

Explanation :

Consider a polynomial function $y=x^4+3x^2+2$ .

The factors of the constant term $2$ are: $\pm 1,\pm 2$ .

The factors of the leading coefficient $1$ are: $\pm 1$ .

So, the possible rational roots of the polynomial function are: $\pm 1,\pm 2$ .

Verify if the possible rational roots are actual rational roots or not using synthetic division.

  $\left\{\begin{array}{l}\overline{)1}\text{ 1 }0302\\ \downarrow 114\text{ 4}\\ \overline{114\text{ 4 }\overline{)6}}\end{array}\right\},\left\{\begin{array}{l}\overline{)-1}\text{ 1 }0302\\ \downarrow -11-4\text{ 4}\\ \overline{1-14-\text{4 }\overline{)6}}\end{array}\right\}$

Here, the remainder is not zero, so $x\ne \pm 1$ .

  $\left\{\begin{array}{l}\overline{)2}\text{ 1 }0302\\ \downarrow 2414\text{ 28}\\ \overline{127\text{ 14 }\overline{)30}}\end{array}\right\},\left\{\begin{array}{l}\overline{)-2}\text{ 1 }0302\\ \downarrow -24-14-\text{28}\\ \overline{1-27-1\text{4 }\overline{)-26}}\end{array}\right\}$

Here, the remainder is not zero, so $x\ne \pm 2$ .

Therefore, the polynomial function $y=x^4+3x^2+2$ has $\pm 1,\pm 2$ as the possible rational roots of the polynomial function but it has no actual rational zeros.