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Zero

A zero (or root) 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 zeroes. 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.

 


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