Final answer:
To find all zeros of a polynomial function the quadratic formula x = (-b ± √(b²-4ac))/(2a) is most commonly used, which provides complex solutions. If b²-4ac > 0, there are two real solutions; if it is 0, there is one real solution; and if it is < 0, there are two complex solutions. Some calculators have functions to aid in finding these roots as well.
Step-by-step explanation:
When you're asked to find all zeros of a polynomial function such as ax²+bx+c = 0, you're looking for the values of x that make the polynomial equal to zero. These are also known as the roots or solutions of the equation. There are several methods for finding these zeros.
The most common method is to use the quadratic formula:
x = (-b ± √(b²-4ac))/(2a)
This formula will give you the complex solutions for any quadratic equation. For real coefficients a, b, and c, if the discriminant (b²-4ac) is positive, you get two distinct real solutions; if it is zero, you get one real solution; and if it is negative, you get two complex solutions.
In case you have access to a TI-83, 83+, or 84 calculator, there is a built-in function to find the polynomial roots directly, which can simplify your work.
Remember that when working with certain problems, especially in subjects like chemistry or physics, you might have to find square roots, cube roots, or other higher roots. Knowing how to perform such operations on your calculator is essential. If you're unsure about how to do this, make sure to reach out to your instructor for assistance.