Answer :
Final answer:
To determine the potential rational zeros of f(x)=6x³-18x²-36x+48, we use the Rational Root Theorem. Any rational zero must be a factor of the constant term over a factor of the leading coefficient. 7 does not fit these criteria, making it not a potential rational zero.
Explanation:
The question asks which is not a potential rational zero of f(x)=6x³-18x²-36x+48. To find the potential rational zeros of a polynomial, we use the Rational Root Theorem, which suggests that any rational zero, expressed as a fraction p/q, must have a numerator (p) that is a factor of the constant term and a denominator (q) that is a factor of the leading coefficient. In this case, the constant term is 48, and the leading coefficient is 6.
The factors of 48 include ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, and ±48, and the factors of 6 include ±1, ±2, ±3, and ±6. Therefore, the potential rational zeros are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, and ±48, and their negatives, all possibly divided by 1, 2, 3, or 6.
A number not fitting this criterion would not be a potential rational zero. 7 is an example that does not meet these criteria, making it not a potential rational zero of the given polynomial.
