Solutions of Polynomials

Question

If two polynomial equations have real solutions, then will the equation that is the result of adding, subtracting, or multiplying the two polynomial equations also have real solutions? Will the resulting equation have complex solutions if one or both of the original equations have complex solutions?

Answer 1

This is probably best understood by graphing. For an equation to have a real solution, the graph has to cross the X-Axis. (What we would graph is an equation where the polynominal is set equal to Y instead of 0)

Adding or subtracting polynomials can result in an equation with either real or imaginary roots. I'm not sure if there are any codicils, but some sample problems will tell us.

One easy example:

The equation X² + 5X + 6 = 0 has solutions at X = -2, -3

The equation X + 4 =0 has a solution at X = -4

Next, add the 2 equations to get:

X² + 6X + 10

This has 2 complex solutions: X = -3 ± i

Try some different combinations.....

MULTIPLYING polynomials is a different story. Each one has either real or imaginary factors. When they are multiplied, the same factors are still there--each one results in the same real or imaginary solutions as in the original.