Final answer:
To find all real and imaginary zeros of a polynomial, determine the function's end behavior, use the Complex Conjugate Root Theorem, the Rational Root Theorem, and perform polynomial division. Understanding these concepts and using a calculator can streamline the process.
Step-by-step explanation:
Finding all real and imaginary zeros for a polynomial function can be a multiple-step process. The function's end behavior is determined by the degree and leading coefficient of the polynomial. For example, as x approaches infinity, a polynomial with a positive leading coefficient and an even degree will also approach infinity. Conversely, with a negative leading coefficient and an even degree, the polynomial will approach negative infinity. Understanding this behavior can guide you on where to look for zeros on the function's graph.
Complex conjugate zeros come into play when dealing with non-real solutions. According to the Complex Conjugate Root Theorem, if a polynomial has real coefficients, any non-real roots must come in conjugate pairs, such as a + bi and a - bi.
The Rational Root Theorem can help to find all possible rational zeros of a polynomial function. With this theorem, you can list all potential rational roots and then use polynomial division or a calculator to verify which, if any, are actual roots of the function.
Polynomial division, either by synthetic division or long division, is used to simplify polynomials and can help in finding the polynomial's zeros. Dividing the polynomial by a binomial root will leave you with a smaller degree polynomial, which should be easier to solve.
It's essential to know how to perform operations like square roots, cube roots, and polynomial division on a calculator. For more complex equations, especially those with non-real roots, a TI-83, 83+, or 84 calculator can be especially useful, offering functions to automate some of the processes involved in finding solutions for quadratic equations like ax² + bx + c = 0.