Question

Explain the difference between finding rational zeros and finding all zeros.

Answer 1

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.

Answer 2

see explanation

Step-by-step explanation:

Given the polynomial

`p(x)=a_nx^n+a_(n-1)x^(n-1)+a_(n-2)x^(n-2)+...+a_2x^2+a_1x+a_o`

The rational zeros are all the possible factors of
`a_0` dived by all the possible factors of
`a_n`. Therefore we use the rational rots theorem to find all possible rational zeros.

To find all zeros we use the remainder theorem. Thus, we plug in all the rational zeros to determine which ones evaluate to zero. These are the real zeros. We could also use long division to find the remaining zeros as soon as we got the first zero with the remainder theorem.