ANSWER
A. -3, -2, 2, and 3
EXPLANATION
The given polynomial function is,
[tex]f(x) = {x}^{4} - 13 {x}^{2} + 36[/tex]
To find the real roots, we equate the function to zero to obtain;
[tex]{x}^{4} - 13 {x}^{2} + 36 = 0[/tex]
We can solve this equation as a quadratic equation in
[tex] {x}^{2}.[/tex]
Thus we rewrite the equation as,
[tex]({x}^{2})^{2} - 13 {x}^{2} + 36 = 0[/tex]
We split the middle term to get,
[tex]({x}^{2})^{2} - 9 {x}^{2} - 4 {x}^{2} + 36 = 0.[/tex]
We factor to get,
[tex] {x}^{2} ( {x}^{2} - 9) - 4( {x}^{2} - 9) = 0[/tex]
We factor to get,
[tex]( {x}^{2} - 9)( {x}^{2} - 4) = 0[/tex]
Either
[tex] {x}^{2} - 9 = 0[/tex]
or
[tex] {x}^{2} - 4 = 0[/tex]
[tex]x = \pm \sqrt{9} \: or \: x = \pm \sqrt{4} [/tex]
[tex]x = \pm3 \: or \: x = \pm2[/tex]
[tex]x=-3,-2,2,3[/tex]
The correct answer is A
