use the rational zeros theorem to list the potential rational zeros of the potential function. Do not attempt to find the zeros.
f(x)=2x^5-x^3+8x^2+52
any rational roots p/q will have p a factor of 52 and q a factor of 2. So, possibilities are
±1 ±2 ±4 ±13 ±26 ±52
±1/2 ±2/2 ±4/2 ±13/2 ±26/2 ±52/2
with some duplicates listed.
To use the rational zeros theorem to list the potential rational zeros of a given function, we need to consider the coefficients of the polynomial. In this case, the function is f(x) = 2x^5 - x^3 + 8x^2 + 52.
1. Identify the constant term: The constant term is 52.
2. Identify the leading coefficient: The leading coefficient is 2.
3. Apply the rational zeros theorem: According to the theorem, the potential rational zeros are all possible combinations of factors of the constant term (52) divided by factors of the leading coefficient (2).
- Factors of 52: ±1, ±2, ±4, ±13, ±26, ±52
- Factors of 2: ±1, ±2
Taking all possible combinations of the factors, we obtain the list of potential rational zeros:
±1/1, ±2/1, ±4/1, ±13/1, ±26/1, ±52/1, ±1/2, ±2/2
After simplifying the fractions, the potential rational zeros are:
±1, ±2, ±4, ±13, ±26, ±52, ±1/2, ±1
Remember, these are potential rational zeros, and we cannot conclude that any or all of these are actual zeros of the function without further analysis or calculation.