Final answer:
To find the potential rational zeros of the polynomial f(x)=21x^4-x^2+49, we use the Rational Root Theorem and consider factors of the constant term (49) and factors of the leading coefficient (21), resulting in the potential zeros ± 1, ± 1/3, ± 7, ± 1/7, and ± 49/3.
Step-by-step explanation:
The question is about finding the potential rational zeros of the polynomial f(x)=21x4-x2+49. To find the potential rational zeros, we use the Rational Root Theorem, which states that if the polynomial has rational zeros, they are of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is 49 and the leading coefficient is 21.
To find the potential rational zeros, we consider the factors of 49 which are ± 1, ± 7, and ± 49, and the factors of 21 which are ± 1, ± 3, ± 7, and ± 21. Thus, the potential rational zeros are ± 1/1, ± 1/3, ± 1/7, ± 1/21, ± 7/1, ± 7/3, ± 7/7, ± 7/21, ± 49/1, ± 49/3, ± 49/7, ± 49/21, which simplifies to ± 1, ± 1/3, ± 7, ± 1/7, and ± 49/3.