Possible and actual rational roots of $$$f{\left(x \right)} = 4 x^{4} - 37 x^{2} + 9$$$
Your Input
Find the rational zeros of $$$4 x^{4} - 37 x^{2} + 9 = 0$$$.
Solution
Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (the coefficient of the constant term) is $$$9$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 3$$$, $$$\pm 9$$$.
These are the possible values for $$$p$$$.
The leading coefficient (the coefficient of the term with the highest degree) is $$$4$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 4$$$.
These are the possible values for $$$q$$$.
Find all possible values of $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{1}{2}$$$, $$$\pm \frac{1}{4}$$$, $$$\pm \frac{3}{1}$$$, $$$\pm \frac{3}{2}$$$, $$$\pm \frac{3}{4}$$$, $$$\pm \frac{9}{1}$$$, $$$\pm \frac{9}{2}$$$, $$$\pm \frac{9}{4}$$$.
Simplify and remove the duplicates (if any).
These are the possible rational roots: $$$\pm 1$$$, $$$\pm \frac{1}{2}$$$, $$$\pm \frac{1}{4}$$$, $$$\pm 3$$$, $$$\pm \frac{3}{2}$$$, $$$\pm \frac{3}{4}$$$, $$$\pm 9$$$, $$$\pm \frac{9}{2}$$$, $$$\pm \frac{9}{4}$$$.
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 $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - 1$$$.
$$$P{\left(1 \right)} = -24$$$; thus, the remainder is $$$-24$$$.
Check $$$-1$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(-1\right) = x + 1$$$.
$$$P{\left(-1 \right)} = -24$$$; thus, the remainder is $$$-24$$$.
Check $$$\frac{1}{2}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \frac{1}{2}$$$.
$$$P{\left(\frac{1}{2} \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$\frac{1}{2}$$$ is a root.
Check $$$- \frac{1}{2}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(- \frac{1}{2}\right) = x + \frac{1}{2}$$$.
$$$P{\left(- \frac{1}{2} \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$- \frac{1}{2}$$$ is a root.
Check $$$\frac{1}{4}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \frac{1}{4}$$$.
$$$P{\left(\frac{1}{4} \right)} = \frac{429}{64}$$$; thus, the remainder is $$$\frac{429}{64}$$$.
Check $$$- \frac{1}{4}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(- \frac{1}{4}\right) = x + \frac{1}{4}$$$.
$$$P{\left(- \frac{1}{4} \right)} = \frac{429}{64}$$$; thus, the remainder is $$$\frac{429}{64}$$$.
Check $$$3$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - 3$$$.
$$$P{\left(3 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$3$$$ is a root.
Check $$$-3$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(-3\right) = x + 3$$$.
$$$P{\left(-3 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$-3$$$ is a root.
Check $$$\frac{3}{2}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \frac{3}{2}$$$.
$$$P{\left(\frac{3}{2} \right)} = -54$$$; thus, the remainder is $$$-54$$$.
Check $$$- \frac{3}{2}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(- \frac{3}{2}\right) = x + \frac{3}{2}$$$.
$$$P{\left(- \frac{3}{2} \right)} = -54$$$; thus, the remainder is $$$-54$$$.
Check $$$\frac{3}{4}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \frac{3}{4}$$$.
$$$P{\left(\frac{3}{4} \right)} = - \frac{675}{64}$$$; thus, the remainder is $$$- \frac{675}{64}$$$.
Check $$$- \frac{3}{4}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(- \frac{3}{4}\right) = x + \frac{3}{4}$$$.
$$$P{\left(- \frac{3}{4} \right)} = - \frac{675}{64}$$$; thus, the remainder is $$$- \frac{675}{64}$$$.
Check $$$9$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - 9$$$.
$$$P{\left(9 \right)} = 23256$$$; thus, the remainder is $$$23256$$$.
Check $$$-9$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(-9\right) = x + 9$$$.
$$$P{\left(-9 \right)} = 23256$$$; thus, the remainder is $$$23256$$$.
Check $$$\frac{9}{2}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \frac{9}{2}$$$.
$$$P{\left(\frac{9}{2} \right)} = 900$$$; thus, the remainder is $$$900$$$.
Check $$$- \frac{9}{2}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(- \frac{9}{2}\right) = x + \frac{9}{2}$$$.
$$$P{\left(- \frac{9}{2} \right)} = 900$$$; thus, the remainder is $$$900$$$.
Check $$$\frac{9}{4}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \frac{9}{4}$$$.
$$$P{\left(\frac{9}{4} \right)} = - \frac{4851}{64}$$$; thus, the remainder is $$$- \frac{4851}{64}$$$.
Check $$$- \frac{9}{4}$$$: divide $$$4 x^{4} - 37 x^{2} + 9$$$ by $$$x - \left(- \frac{9}{4}\right) = x + \frac{9}{4}$$$.
$$$P{\left(- \frac{9}{4} \right)} = - \frac{4851}{64}$$$; thus, the remainder is $$$- \frac{4851}{64}$$$.
Answer
Possible rational roots: $$$\pm 1$$$, $$$\pm \frac{1}{2}$$$, $$$\pm \frac{1}{4}$$$, $$$\pm 3$$$, $$$\pm \frac{3}{2}$$$, $$$\pm \frac{3}{4}$$$, $$$\pm 9$$$, $$$\pm \frac{9}{2}$$$, $$$\pm \frac{9}{4}$$$A.
Actual rational roots: $$$\frac{1}{2}$$$, $$$- \frac{1}{2}$$$, $$$3$$$, $$$-3$$$A.