Final answer:
The difference between finding rational zeros and finding all zeros lies in the characteristics of the solutions. Rational zeros are solutions that can be expressed as a ratio of two integers, and they can be obtained using the Rational Root Theorem. Finding all zeros involves determining all of the complex and real solutions to a polynomial equation, regardless of whether they are rational or irrational.
Step-by-step explanation:
The difference between finding rational zeros and finding all zeros lies in the characteristics of the solutions. Rational zeros, also known as rational roots, are solutions to a polynomial equation that can be expressed as a ratio of two integers. These zeros can be obtained using the Rational Root Theorem, which provides a method for finding potential rational solutions. On the other hand, finding all zeros involves determining all of the complex and real solutions to a polynomial equation, regardless of whether they are rational or irrational.
For example, consider the polynomial equation f(x) = x^3 - 4x^2 + 5x - 2. Using the Rational Root Theorem, we can find that the potential rational zeros are ±1 and ±2. By testing these values in the equation, we can determine that the rational zeros are x = 1 and x = 2. However, there may be additional zeros that are not rational, which would require further investigation or the use of other methods such as factoring or the quadratic formula.
In summary, finding rational zeros focuses specifically on determining solutions that can be expressed as ratios of integers, while finding all zeros involves identifying all possible solutions, including rational and irrational numbers.