Question

solving polynomial equations using technology
examining a limitation of a method of solving polynomial equations
a student graphs two polynomial equations and states that since they do not intersect, the system has no solution. is the student correct?
the student is incorrect because the systems could have only complex solutions that cannot be found by graphing
the student’s reasoning is correct.
the student is incorrect because the solutions are the x - intercepts of the graphs.
the student is incorrect because the systems could have only irrational solutions that cannot be found by graphing.

Explanation:

To determine if the student is correct, we analyze the nature of polynomial systems. A system of polynomial equations' solutions are the intersection points of their graphs. However, even if the graphs (as functions) don't intersect, the system could have complex solutions (solutions with imaginary parts) that aren't visible in the real - plane graphing. The student assumes no intersection (in real - plane) means no solution, but complex solutions are possible. So the first option (stating the student is incorrect because the systems could have only complex solutions that cannot be found by graphing) is correct. The student's reasoning is wrong because graphing (in real - plane) can't show complex solutions, and a system of polynomials can have complex solutions even if their real - valued graphs don't intersect.

Answer:

The student is incorrect because the systems could have only complex solutions that cannot be found by graphing.