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Possible and actual rational roots of $$$f{\left(x \right)} = x^{4} + 50 x^{2} + 625$$$
Your Input
Find the rational zeros of $$$x^{4} + 50 x^{2} + 625 = 0$$$.
Solution
Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (the coefficient of the constant term) is $$$625$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 5$$$, $$$\pm 25$$$, $$$\pm 125$$$, $$$\pm 625$$$.
These are the possible values for $$$p$$$.
The leading coefficient (the coefficient of the term with the highest degree) is $$$1$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$.
These are the possible values for $$$q$$$.
Find all possible values of $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{5}{1}$$$, $$$\pm \frac{25}{1}$$$, $$$\pm \frac{125}{1}$$$, $$$\pm \frac{625}{1}$$$.
Simplify and remove the duplicates (if any).
These are the possible rational roots: $$$\pm 1$$$, $$$\pm 5$$$, $$$\pm 25$$$, $$$\pm 125$$$, $$$\pm 625$$$.
Next, check the possible roots: if $$$a$$$ is a root of the polynomial $$$P{\left(x \right)}$$$, the remainder from the division of $$$P{\left(x \right)}$$$ by $$$x - a$$$ should equal $$$0$$$ (according to the remainder theorem, this means that $$$P{\left(a \right)} = 0$$$).
Check $$$1$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - 1$$$.
$$$P{\left(1 \right)} = 676$$$; thus, the remainder is $$$676$$$.
Check $$$-1$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - \left(-1\right) = x + 1$$$.
$$$P{\left(-1 \right)} = 676$$$; thus, the remainder is $$$676$$$.
Check $$$5$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - 5$$$.
$$$P{\left(5 \right)} = 2500$$$; thus, the remainder is $$$2500$$$.
Check $$$-5$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - \left(-5\right) = x + 5$$$.
$$$P{\left(-5 \right)} = 2500$$$; thus, the remainder is $$$2500$$$.
Check $$$25$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - 25$$$.
$$$P{\left(25 \right)} = 422500$$$; thus, the remainder is $$$422500$$$.
Check $$$-25$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - \left(-25\right) = x + 25$$$.
$$$P{\left(-25 \right)} = 422500$$$; thus, the remainder is $$$422500$$$.
Check $$$125$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - 125$$$.
$$$P{\left(125 \right)} = 244922500$$$; thus, the remainder is $$$244922500$$$.
Check $$$-125$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - \left(-125\right) = x + 125$$$.
$$$P{\left(-125 \right)} = 244922500$$$; thus, the remainder is $$$244922500$$$.
Check $$$625$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - 625$$$.
$$$P{\left(625 \right)} = 152607422500$$$; thus, the remainder is $$$152607422500$$$.
Check $$$-625$$$: divide $$$x^{4} + 50 x^{2} + 625$$$ by $$$x - \left(-625\right) = x + 625$$$.
$$$P{\left(-625 \right)} = 152607422500$$$; thus, the remainder is $$$152607422500$$$.
Answer
Possible rational roots: $$$\pm 1$$$, $$$\pm 5$$$, $$$\pm 25$$$, $$$\pm 125$$$, $$$\pm 625$$$A.
Actual rational roots: no rational roots.