The potential roots of the given function can be found in plus/minus 2 and plus/minus 3.
The given function;
[tex]p(x) = x^4 -9x^2 -4x + 12[/tex]
A potential root of the given function must be a zero of the function.
To determined the zeros of the function, we are going to test each of the given values and determine the values that make our function to be zero.
Test for +/-2;
[tex]p(2) = (2)^4 -9(2)^2 - 4(2)+ 12= -16\\\\p(-2) = (-2)^4 - 9(-2)^2 -4(-2) + 12= 0[/tex]
Test for +/-4;
[tex]p(4) = (4)^4 -9(4)^2 -4(4) + 12 = 108\\\\p(-4) = (-4)^4 -9(-4)^2 -4(-4) + 12 =140[/tex]
Test for +/-9;
[tex]p(9) = (9)^4 - 9(9)^2 -4(9)+ 12= 5820\\\\p(-9) = (-9)^4 - 9(-9)^2 -4(-9)+ 12=5880[/tex]
Test for +/- 0.5;
[tex]p(0.5) = (0.5)^4 -9(0.5)^2 -4(0.5) + 12 = 7.81\\\\p(-0.5) = (-0.5)^4 -9(-0.5)^2 -4(-0.5) + 12= 11.81[/tex]
Test for +/- 3;
[tex]p(3) = (3)^4 -9(3)^2 -4(3) + 12= 0\\\\p(-3) = (-3)^4 -9(-3)^2 -4(-3) + 12= 24[/tex]
Test for +/- 6;
[tex]p(6) = (6)^4 -9(6)^2 -4(6) + 12 = 960\\\\p(-6) = (-6)^4 -9(-6)^2 -4(-6) + 12 = 1008[/tex]
Test for +/- 12;
[tex]p(12) = (12)^4 -9(12)^2 -4(12) + 12 = 19404\\\\p(-12) = (-12)^4 -9(-12)^2 -4(-12) + 12 =19500[/tex]
Thus, the potential roots of the given function can be found in the following options;
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