Final answer:
To evaluate if the equation 12x^4-x^2-6 has no rational root, we apply the Rational Root Theorem, which provides a list of potential rational roots based on the factors of the constant term and the leading coefficient. After testing all possible rational roots and finding none that satisfy the equation, we can conclude that the equation has no rational roots.
Step-by-step explanation:
To show that the equation 12x^4-x^2-6 has no rational root, we can use the Rational Root Theorem. This theorem states that for a polynomial equation of the form ax^n + bx^(n-1) + ... + zx + y = 0, where all coefficients are integers, any rational solution p/q (where p and q are coprime integers) must have p as a factor of the constant term (y) and q as a factor of the leading coefficient (a).
In our case, the constant term is -6, whose factors are ±1, ±2, ±3, and ±6. The leading coefficient is 12, with factors ±1, ±2, ±3, ±4, ±6, and ±12. Thus, any rational root of the equation would have to be in the form of ± (1, 2, 3, 6) / (1, 2, 3, 4, 6, 12).
We can test each possible rational root by plugging them into the equation. If none of the possible rational roots satisfies the equation, we can conclude that there are no rational roots. This involves a bit of computation, as you will need to substitute each candidate and see if it results in a zero value for the polynomial.
After testing all possibilities, if no rational root is found, we have shown that the equation has no rational root. It's important to note that while the equation might still have real roots, they would be irrational or complex, and proving that would require other methods such as graphing, completing the square, or using the quadratic formula.