Why nth Degree Equations Have n Roots?

[johncena]

Why is it said that an equation of nth degree must possesses n roots ?
if x^1 = y, x has only 1 value
x^2 = y, x has 2 values (the 2 values may be equal)
x^3 = y, x has 3 values
going on like this, we have, x^0 = 1 , implies x has no solutions. but x has infinite number of solutions.

 

[rasmhop]

What people say when they say that an equation of degree n has n roots is that given a polynomial P(x) of degree n, it has n complex roots (where n is a non-negative integer). However x^0=1 so the polynomial in your last example is:
P(x) = x^0 - 1 = 1-1 = 0
This does not have degree 0. We often say that it has degree $-\infty$, but since it's the only polynomial with degree not a non-negative integer this is the single polynomial to which the rule does not apply.

 

[ABarrios]

Fundamental Theorem of Algebra: Every polynomial of degree n (n=/=0) has exactly n roots counting multiplicity over the Complex numbers. In the case of the real numbers, it has d roots where d is less than or equal to n.
For instance, the easiest example x^2+1=0 only has complex solutions, namely i and -i, but over the Reals, it has no roots.
Now to finish your question consider nonzero polynomials of degree 0, suggesting they are nonzero, they have no solutions. E.g., f(x)=5 is a polynomial of degree 0 and has 0 roots.

 

[g_edgar]

I guess $x^{1/2} = 4$ has half a solution!

 

[eumyang]

I guess $x^{1/2} = 4$ has half a solution!

I'm assuming that you're joking, as this would not be a polynomial in the first place. The Fundamental Theorem of Algebra applies to non-constant single-variable polynomials with complex coefficients.

I don't know why, but finding nth roots of complex numbers is one of my favorite topics in teaching Pre-Calculus. I find it fascinating to see that you can find nth roots algebraically (like, for instance solving the equation $$x^{4}-1=0$$ to find the fourth roots of unity), or by using polar form ($$1 = cos 0 + i sin 0$$) and get the same answers. I usually get a 'wow' moment from my students when I show them this.


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