Question

In Problems 21–32, determine the maximum number of real zeros that each polynomial function may have. Then list the potential rational zeros of each polynomial function. Do not attempt to find the zeros.

$f\left(x\right)=x^5-x^4+2x^2+3$

Short answer

The given polynomial has maximum of $5$real zeros.

The potential rational zeros are $\pm 1,\pm 3$.

Step-by-step solution

Step 1. Given Information 

We are given a function,

$f\left(x\right)=x^5-x^4+2x^2+3$

We will use Rational zeroes theorem to find the potential zeros of given polynomial.

Step 2. Finding the maximum number of real zeros

The degree of given polynomial is $5$,so the maximum number of real zeros is localid="1646107202708" $5$.

Step 3. Listing the potential rational zeros   

Now, using the Rational zeroes theorem, we get

$$\begin{gathered} a_0=3 \\ {a}_5=1 \end{gathered}$$

All the integers p that are factors of $a_0=3$ and integers q that are the factors of $a_5=1$are,

$$\begin{gathered} p=\pm 1,\pm 3 \\ q=1 \end{gathered}$$

So, the potential rational zeros are,

$\frac{p}{q}=\pm 1,\pm 3$

Hence if the function have rational zeros then it will found in the above given list which contains four possibilities.