Answer
The polynomial function \[{{x}^{4}}+6{{x}^{2}}+2=0\] does not have any rational roots.
Work Step by Step
According to the Rational Zero Theorem,
Possible rational zeros = (Factors of the constant term)/ (Factors of the leading coefficient) ... (2)
The constant term in equation (1) is 2.
All the factors of 2 are $\pm 1\,\text{ and }\pm 2.$
The leading coefficient of the given polynomial in equation (1) is $+1$.
Factors of the constant term 4 are $\pm 1\,\text{ and }\pm 2$.
Factors of the leading coefficient are 1 and $\pm 1.$
Using the formula for the possible rational zeroes, we get that the possible rational zeroes are $\pm 1,\pm 2$.
But from equation (1), we get that the minimum value of the polynomial is 2 and the terms have even power so on plotting the curve it will not intersect the x axis and it will always lie in the +x- axis.
Hence, the polynomial function ${{x}^{4}}+6{{x}^{2}}+2=0$ will not have any rational roots.
