Zero
A zero of a function is an intersection point or touch point with the x-axis.
Explanation
A function may have none, one, or a finite number or infinitely many roots. You can write this as f (x) = 0.
Exemple 1
For the intersection point of the linear function f (x) = 3x + 6 with the x-axis applies
3x + 6 = 0
The root is x = −2.
Example 2
For the intersection points of the quadratic function f (x) = x2 + 2x − 15 with the x-axis applies
x2 + 2x − 15 = 0
Factorization gives (x − 3) (x + 5) = 0 so that the two zeroes are at x = 3 and x = −5.
Example 3
For the intersection points of a polynomial in one variable z with the x-axis applies
a0 + a1z + a2z2 + ··· + anzn = 0
Factorization in linear factors yields an(z − b1)(z − b2) ··· (z − bn) = 0 so that there are n zeroes.
Example 4
For the intersection points of the sine function f (x) = sin (x) with the x-axis applies
sin (x) = 0
There are infinitely many zeroes, because sin (0) = sin (π) = sin (2π) = 0 etc.