What are function zeros? The answer is quite simple - this is a mathematical term, which means the domain of definition of a given function on which its value is zero. Zeros of a function are also called the roots of the equation. The easiest way to explain what function zeros are is with a few simple examples.
Examples
We consider the simple equation y = x + 3. Since the zero of the function is the value of the argument at which y has acquired a zero value, we substitute 0 in the left side of the equation:
0 = x + 3;
x = -3.
In this case, -3 is the desired zero. For this function, there is only one root of the equation, but this is not always the case.
Consider another example:
y = x 2 -9.
Substitute 0 in the left side of the equation, as in the previous example:
0 = x 2 -9;
-9 = x 2 .
Obviously, in this case there will be two zeros of the function: x = 3 and x = -3. If the equation had an argument of the third degree, there would be three zeros. We can make a simple conclusion that the number of roots of the polynomial corresponds to the maximum degree of agrument in the equation. However, many functions, for example, y = x 3 , at first glance contradict this statement. Logic and common sense suggest that this function has only one zero - at the point x = 0. But actually there are three roots, they just all coincide. If you solve the equation in complex form, it becomes obvious. x = 0 in this case, the root whose multiplicity is 3. In the previous example, the zeros did not coincide, therefore they had a multiplicity of 1.
Definition Algorithm
The examples presented show how to determine the zeros of a function. The algorithm is always the same:
- Write function.
- Substitute y or f (x) = 0.
- Solve the resulting equation.
The complexity of the last paragraph depends on the degree of the argument of the equation. When solving equations of high degrees, it is especially important to remember that the number of roots of the equation is equal to the maximum degree of the argument. This is especially true for trigonometric equations, where the division of both parts by sine or cosine leads to loss of roots.
Equations of an arbitrary degree are most easily solved by the Horner method, which was developed specifically for finding zeros of an arbitrary polynomial.
The value of zeros of functions can be either negative or positive, real or lying in the complex plane, single or multiple. Or the roots of the equation may not be. For example, the function y = 8 will not get a zero value for any x, because it does not depend on this variable.
The equation y = x 2 -16 has two roots, and both lie in the complex plane: x 1 = 4Ρ, x 2 = -4Ρ.
Common mistakes
A common mistake made by students who have not yet figured out exactly what function zeros are is to replace the argument (x) with zero, and not the value (y) of the function. They confidently substitute x = 0 in the equation and, based on this, find y. But this is the wrong approach.
Another mistake, as already mentioned, is the reduction by sine or cosine in the trigonometric equation, due to which one or more zeros of the function are lost. This does not mean that nothing can be reduced in such equations, itβs just that in future calculations it is necessary to take these βlostβ factors into account.
Graphical representation
Understanding what zeros of a function is is possible using mathematical programs such as Maple. In it, you can build a graph by indicating the desired number of points and the desired scale. Those points at which the graph crosses the OX axis are the desired zeros. This is one of the fastest ways to find the roots of a polynomial, especially if its order is higher than the third. So if there is a need to regularly perform mathematical calculations, find the roots of polynomials of arbitrary degrees, build graphs, Maple or a similar program will be simply indispensable for the implementation and verification of calculations.