Answer :

To find the possible rational zeros of the polynomial [tex]\( n(x) = -4x^5 + 19x^3 + 2x^2 - 10 \)[/tex], we can use the Rational Root Theorem. This theorem suggests that any rational root, expressed in its lowest terms [tex]\( \frac{p}{q} \)[/tex], must have [tex]\( p \)[/tex] as a factor of the constant term and [tex]\( q \)[/tex] as a factor of the leading coefficient.

Here’s how to find the factors:

1. Identify the constant term and the leading coefficient:
- Constant term = [tex]\(-10\)[/tex]
- Leading coefficient = [tex]\(-4\)[/tex]

2. List the factors of the constant term ([tex]\(-10\)[/tex]):
- [tex]\(\pm 1, \pm 2, \pm 5, \pm 10\)[/tex]

3. List the factors of the leading coefficient ([tex]\(-4\)[/tex]):
- [tex]\(\pm 1, \pm 2, \pm 4\)[/tex]

4. Form all possible combinations of [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient:
- Possible rational zeros can be: [tex]\(\pm 1, \pm 1/2, \pm 1/4, \pm 2, \pm 5, \pm 5/2, \pm 5/4, \pm 10, \pm 10/2, \pm 10/4\)[/tex]

5. Simplify the fractions:
- Simplified possible rational zeros: [tex]\(\pm 1, \pm 1/2, \pm 1/4, \pm 2, \pm 5, \pm 5/2, \pm 5/4, \pm 10\)[/tex]

However, upon further inspection, there are no rational roots that satisfy the polynomial equation. Therefore, in this case, the final conclusion is that there are no rational zeros for the polynomial [tex]\( n(x) = -4x^5 + 19x^3 + 2x^2 - 10 \)[/tex].

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Rewritten by : Barada