What is the difference between a zero and a root?

Question

This is an Algebra 2 question. If anyone knows, thank you in advance!!

Answer 1

A zero can be many things, including a root.

For example, a zero (in a field) is any value which, when multiplied by any other value in the field, returns itself, and when added, returns the other value.

Let x be any value in a field.
Let z be a zero in that field.

Then z*x = z (for any x),
and z+x = x (for any x).

In real numbers (and in complex numbers), the 'zero' is, of course, 0.

0*x = 0 (for any x in real or complex numbers)

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root of an equation: a number which, when substituted for the variable in the equation, reduces it to an identity. (Mathematics Dictionary, by G.James and R.C.James)
A polynomial is an addition of terms, therefore the identity is 0. The roots of a polynomial are the values that give a total of zero for the equation.

(An identity leaves the other object unchanged. In real numbers, when adding, 0 is the identity because x + 0 = x for any x. When multiplying, 1 is the identity because 1*x = x for any x)

However, there are equations where the identity is not zero. So the root is not always a 'zero'. But that is beyond Algebra 2.
At Algebra two level, the roots of an equation means the same as the zeroes of an equation.

If you want a subtle difference:

take f(x) = (x-a)^2

There is one 'zero', that is when x = a.

There are two roots (because it is a second degree polynomial) and BOTH roots are x = a.
This is called a multiple root (in this case: a double root)

A multiple root implies that x=a will also be a root of the derivative f'(x).


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A root can also mean a number such that when multiplied by itself a given number of times, will return the original value.

For example, the 12th root of 2 is 1.059463...
This is the ratio of frequencies between two consecutive half-tones on a modern piano (e.g., from C to C-sharp); there are 12 half-tones in an octave.

Answer 2

Well from what I know, for instance the equation:

x(squared) - 1
when you factor it you get two roots, (x + 1)(x - 1) and to find out the actual roots you have to set them equal to zero. Like x + 1 = 0 and x - 1 = 0, then solve. I don't really know what you were trying to ask.