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another way to present the fundamental theorem of algebra is to state that, "for any polynomial equation of degree n with real coefficients, there exist n complex solutions." why is it important to learn about complex solutions? why is it remarkable that with real coefficients, there must be complex solutions?

answer:

1. imaginary numbers, also called complex numbers, are used in real-life applications, such as electricity, as well as quadratic equations. ac electricity changes between positive and negative in a sine wave. combining ac currents can be very difficult because they may not match properly on the waves.

2. a polynomial of degree n can have only an even number fewer than n real roots. thus, when we count multiplicity, a cubic polynomial can have only three roots or one root; a quadratic polynomial can have only two roots or zero roots.

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