Root of a function facts for kids

A root of a function is a special number. When you put this number into the function, the answer you get is zero. Think of it like a puzzle: what number makes the function equal to 0?

For example, if you have a function like f(x) = x - 5, the root is 5 because if you put 5 in, f(5) = 5 - 5 = 0.

What is a Root of a Function?

A root of a function is also called a zero of the function. It's the input value (often called x) that makes the function's output (often called f(x)) exactly zero.

In math, we write this as: f(x) = 0.

Imagine you have a graph of a function. The roots are the points where the graph crosses or touches the horizontal line called the x-axis. At these points, the y-value (which is f(x)) is always zero.

Why are Roots Important?

Roots are very useful in many areas of math and science. They help us solve problems like:

• Finding when something reaches a certain level (like zero).
• Understanding where a graph crosses the x-axis.
• Solving equations.

Roots of Polynomials

A polynomial is a special type of function. It's made up of terms added together, where each term is a number multiplied by a variable raised to a whole number power (like x², x³, etc.). For example, f(x) = 2x + 3 or g(x) = x² - 4 are polynomials.

Because polynomials are also functions, they have roots too. These are often called polynomial roots.

The Fundamental Theorem of Algebra

There's a very important rule in math called the fundamental theorem of algebra. It tells us something cool about the roots of polynomials.

This theorem says that every polynomial with complex numbers as its coefficients (the numbers multiplied by the variables) has at least one root that is a complex number.

What are complex numbers? They are numbers that can be written as a + bi, where a and b are regular numbers, and i is the imaginary unit (where i² = -1). Don't worry too much about them for now, just know they include all the numbers you usually use, plus some new ones.

How Many Roots Can a Polynomial Have?

The fundamental theorem of algebra, along with another rule called the polynomial remainder theorem, helps us figure out how many roots a polynomial has.

Every complex polynomial of degree n has exactly n complex roots.

What does "degree n" mean? The degree of a polynomial is the highest power of the variable in the polynomial.

• For example, in f(x) = 2x + 3, the highest power of x is 1, so its degree is 1. It has 1 root.
• In g(x) = x² - 4, the highest power of x is 2, so its degree is 2. It has 2 roots.

Sometimes, some of these roots might be the same. For example, the polynomial h(x) = x² - 2x + 1 has a degree of 2. It has two roots, but both of them are 1. We say the root 1 has a "multiplicity" of 2.