Zero of a function facts for kids
In mathematics, a zero (also called a root) of a function is a special number that makes the function's output equal to zero. Imagine you have a function, like a machine that takes a number as input and gives you another number as output. A zero is the input number that makes the machine output exactly 0.
For example, if you have a function `f(x) = x - 5`, the zero is 5, because when you put 5 into the function, you get `5 - 5 = 0`.
When we talk about functions that map real numbers to real numbers, their zeros are easy to spot on a graph. They are the points where the graph crosses or touches the x-axis. These points are also called x-intercepts.
A root of a polynomial is just a zero of that polynomial function. For example, the polynomial `f(x) = x^2 - 5x + 6` has two roots: 2 and 3.
- If you put 2 into the function: `2^2 - (5 × 2) + 6 = 4 - 10 + 6 = 0`.
- If you put 3 into the function: `3^2 - (5 × 3) + 6 = 9 - 15 + 6 = 0`.
Both 2 and 3 make the function equal to 0, so they are its roots or zeros.
Contents
Finding Equation Solutions
Every equation can be rewritten so that all its parts are on one side, and the other side is 0. For example, if you have the equation `x + 3 = 7`, you can rewrite it as `x + 3 - 7 = 0`, which simplifies to `x - 4 = 0`. Now, this looks like a function `f(x) = x - 4` set to zero. So, finding the solutions to an equation is the same as finding the zeros of a function! The solution to `x - 4 = 0` is `x = 4`, which is also the zero of the function `f(x) = x - 4`.
Polynomial Roots Explained
Polynomials are special types of functions.
- A polynomial with an odd degree (like `x^3` or `x^5`) will always have at least one real root. This is because their graphs go from negative infinity to positive infinity (or vice versa), so they must cross the x-axis at least once.
- A polynomial with an even degree (like `x^2` or `x^4`) might not have any real roots. For example, `f(x) = x^2 + 1` never crosses the x-axis because `x^2` is always 0 or positive, so `x^2 + 1` is always 1 or greater.
The Fundamental Theorem of Algebra
This is a very important theorem in mathematics! It tells us that a polynomial of degree `n` (where `n` is the highest power of `x` in the polynomial) will always have exactly `n` roots. However, these roots can be complex numbers (numbers that include the imaginary unit `i`, where `i^2 = -1`), and some roots might be repeated (called "multiplicities"). For polynomials with real numbers as coefficients, any complex roots always come in pairs. This means if `a + bi` is a root, then `a - bi` will also be a root.
How to Compute Roots
Finding the roots of functions, especially polynomials, can be tricky!
- For simple polynomials, like those with a degree of 1 (e.g., `x - 5 = 0`) or 2 (e.g., `x^2 - 4 = 0`), you can often find the roots using basic algebra or the quadratic formula.
- For polynomials up to degree 4, there are also specific formulas to find their roots.
- For polynomials of degree 5 or higher, there are no general formulas that use only basic arithmetic operations and roots (like square roots or cube roots). For these, and for most other functions, mathematicians use special approximation techniques like Newton's method. These methods help find a very close estimate of the root.
Zero Set
In more advanced mathematics, the zero set of a function is simply the collection of all its zeros. It's like a list of every input number that makes the function's output equal to 0.
For example, if `f(x) = x^2 - 4`, its zeros are -2 and 2. So, its zero set would be `{-2, 2}`.
Real-World Applications
Zero sets are used in many areas of mathematics:
- In algebraic geometry, they help define shapes and spaces using polynomial equations.
- In differential geometry, they are used to describe manifolds, which are shapes that look like flat spaces when you zoom in very close (like the surface of a sphere). For example, a sphere is the zero set of the function `f(x,y,z) = x^2 + y^2 + z^2 - 1`.
See also
- Marden's theorem
- Root-finding algorithm
- Sendov's conjecture
- Zero crossing
- Zeros and poles
