Let f(x) be polynomial function. Explain in your own words how to use the Rational Zero Theorem to find the real zeros of f (x).
if any rational number p/q is a zero, p must be a factor of the constant term, and q must be a factor of the coefficient of the highest power of the variable.
To use the Rational Zero Theorem to find the real zeros of a polynomial function f(x), you need to follow certain steps.
Step 1: Determine the leading coefficient and the constant term of the polynomial function f(x). The leading coefficient is the coefficient of the highest degree term, and the constant term is the coefficient of the term with no variable.
Step 2: Write down all the possible rational zeros of the polynomial function. These are the possible values of x that could make the function equal to zero. The Rational Zero Theorem states that if there is a rational zero of a polynomial function, it must be of the form p/q, where p is a factor of the constant term, and q is a factor of the leading coefficient.
Step 3: Use synthetic division or long division to test each possible rational zero. Start with one of the possible zeros and divide the function by (x - possible zero). If the remainder is zero, then that possible zero is a real zero of the polynomial function. If the remainder is not zero, try the next possible zero until you find a real zero.
Step 4: Once you find a real zero, you can then use the Factor Theorem to factor the polynomial function. Divide the polynomial function by (x - real zero) using synthetic division or long division. The resulting quotient will be a lower degree polynomial. Repeat this process with the quotient until you cannot divide any further. The factors obtained from this process represent the factors of the original polynomial.
Step 5: Solve for the remaining zeros of the polynomial by equating the factored form to zero and solving for x. These zeros may be real or complex, depending on the coefficients of the polynomial.
By following these steps, you can use the Rational Zero Theorem to systematically find the real zeros of a polynomial function f(x).