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Possible and actual rational roots of $$$f{\left(x \right)} = x^{4} - 48 x^{2} - 49$$$
Your Input
Find the rational zeros of $$$x^{4} - 48 x^{2} - 49 = 0$$$.
Solution
Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (the coefficient of the constant term) is $$$-49$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 7$$$, $$$\pm 49$$$.
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 $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{7}{1}$$$, $$$\pm \frac{49}{1}$$$.
Simplify and remove the duplicates (if any).
These are the possible rational roots: $$$\pm 1$$$, $$$\pm 7$$$, $$$\pm 49$$$.
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 $$$x^{4} - 48 x^{2} - 49$$$ by $$$x - 1$$$.
$$$P{\left(1 \right)} = -96$$$; thus, the remainder is $$$-96$$$.
Check $$$-1$$$: divide $$$x^{4} - 48 x^{2} - 49$$$ by $$$x - \left(-1\right) = x + 1$$$.
$$$P{\left(-1 \right)} = -96$$$; thus, the remainder is $$$-96$$$.
Check $$$7$$$: divide $$$x^{4} - 48 x^{2} - 49$$$ by $$$x - 7$$$.
$$$P{\left(7 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$7$$$ is a root.
Check $$$-7$$$: divide $$$x^{4} - 48 x^{2} - 49$$$ by $$$x - \left(-7\right) = x + 7$$$.
$$$P{\left(-7 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$-7$$$ is a root.
Check $$$49$$$: divide $$$x^{4} - 48 x^{2} - 49$$$ by $$$x - 49$$$.
$$$P{\left(49 \right)} = 5649504$$$; thus, the remainder is $$$5649504$$$.
Check $$$-49$$$: divide $$$x^{4} - 48 x^{2} - 49$$$ by $$$x - \left(-49\right) = x + 49$$$.
$$$P{\left(-49 \right)} = 5649504$$$; thus, the remainder is $$$5649504$$$.
Answer
Possible rational roots: $$$\pm 1$$$, $$$\pm 7$$$, $$$\pm 49$$$A.
Actual rational roots: $$$7$$$, $$$-7$$$A.