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How is the degree of a polynomial related to the concept of complex numbers?
can someone help me with this Its hard for me to understand
2 answers
The degree of the polynomial determines how many roots the corresponding
polynomial equation has.
Some my be real and some may be complex
If the degree is odd, it must have at least 1 real root.
eg. x^3 + 8 = 0 , it will have 3 roots
we can factor it:
(x+2)(x^2 - 2x + 4) = 0
x = -2, there is your real root
x^2 - 2x + 4 = 0
x = (2 ± √-12)/2 = 1 ± √3 i , so there are the 3 roots
If there are complex roots, they will always come in conjugate pairs, see my example
we could have all complex roots
e.g. x^4 + 16 = 0
there should be 4 roots
x^4 = -16
x^2 = ± 4 i
x = ±(± 2√i)
At this point, don't worry too much about what √i means or is, it would involve
a concept called De Moivre's Theorem
polynomial equation has.
Some my be real and some may be complex
If the degree is odd, it must have at least 1 real root.
eg. x^3 + 8 = 0 , it will have 3 roots
we can factor it:
(x+2)(x^2 - 2x + 4) = 0
x = -2, there is your real root
x^2 - 2x + 4 = 0
x = (2 ± √-12)/2 = 1 ± √3 i , so there are the 3 roots
If there are complex roots, they will always come in conjugate pairs, see my example
we could have all complex roots
e.g. x^4 + 16 = 0
there should be 4 roots
x^4 = -16
x^2 = ± 4 i
x = ±(± 2√i)
At this point, don't worry too much about what √i means or is, it would involve
a concept called De Moivre's Theorem