Explain why the graphing calculator cannot be used to solve or approximate solutions to all polynomial equations.
Did your explanation include the following?
Some polynomial equations have complex solutions.
Therefore, complex solutions cannot be found by graphing the related system or function in the Cartesian (x-y) plane.
Polynomial functions with all complex solutions do not have x-intercepts, and the related system of equations will not intersect.
All polynomials cannot be solved or approximated using graphing calculators because some polynomials have complex solutions.
Take for instance:
\(\mathbf{P(x) = x^3 - 5x^2 + 6x}\)
The above polynomial has real roots, and can be solved and approximated using graphing calculators.
The roots are: 0, 2 and 3
However, the following polynomial cannot be solved using graphing calculators.
\(\mathbf{P(x) =x^3 + 10x^2 + 169x}\)
This is so, because the polynomial has complex solutions.
Hence, some polynomials cannot be solved using graphs because they have complex roots.
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