Answer: the possible rational zeros for f(x) = 5x^2 - 21x + 4 are ±1/5, ±1/4, ±2/5, and ±1/2.
Please note that the Rational Root Theorem provides only the possible rational zeros. To determine the actual zeros of the functions, further calculations or methods, such as factoring or the quadratic formula, may be required.
Step-by-step explanation: To find the possible rational zeros of a polynomial function, we can use the Rational Root Theorem. According to this theorem, the rational zeros are all the possible values that can be obtained by dividing a factor of the constant term by a factor of the leading coefficient.
Let's find the possible rational zeros for each function:
1. For the function f(x) = x^2 - 7x - 10:
- The constant term is -10, and its factors are ±1, ±2, ±5, and ±10.
- The leading coefficient is 1, and its factors are ±1.
- Therefore, the possible rational zeros are the fractions that can be obtained by dividing a factor of -10 by a factor of 1. These include ±1, ±2, ±5, and ±10.
So, the possible rational zeros for f(x) = x^2 - 7x - 10 are ±1, ±2, ±5, and ±10.
2. For the function f(x) = 5x^2 - 21x + 4:
- The constant term is 4, and its factors are ±1, ±2, and ±4.
- The leading coefficient is 5, and its factors are ±1 and ±5.
- Therefore, the possible rational zeros are the fractions that can be obtained by dividing a factor of 4 by a factor of 5. These include ±1/5, ±1/4, ±2/5, and ±2/4 (which simplifies to ±1/2).
So, the possible rational zeros for f(x) = 5x^2 - 21x + 4 are ±1/5, ±1/4, ±2/5, and ±1/2.