Identify coefficients, use Rational Zeros Theorem to list potential zeros, test with synthetic or long division. If the result is zero, it's a real zero. Repeat for all. Account for possible irrational or complex zeros.
To find all real zeros of a polynomial function using the Rational Zeros Theorem, start by identifying the coefficients of the polynomial.
If the polynomial is of degree n with coefficients in the form an*x^n + a(n-1)x^(n-1) + ... + a1x + a0, the theorem states that any rational zero (if it exists) is of the form p/q, where p is a factor of the constant term a0 and q is a factor of the leading coefficient an.
List all possible rational zeros using these factors, considering both positive and negative values. Use synthetic division or polynomial long division to test each potential zero.
If the result is zero, then that value is a real zero of the function. Repeat the process until all real zeros are identified. Note that some zeros may be irrational or complex, so further techniques may be needed to find those.