Fundamental theorem of algebra Facts for Kids

The fundamental theorem of algebra is a very important rule in mathematics. It helps us understand polynomials. A polynomial is a special kind of math expression, like Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 + 3x - 5 . This theorem tells us how many answers, or "roots," a polynomial equation can have. It was first proven by the famous German mathematician Carl Friedrich Gauss.

This theorem says that for any polynomial equation like $f(x)=0$ , if the highest power of $x$ in it is $n$ (and $n$ is greater than zero), then there must be at least one answer for $x$ . In fact, if you count all types of answers, there will be exactly $n$ answers!

What is a Polynomial?
Understanding the Degree of a Polynomial
What are Polynomial Roots?
Counting All the Roots
- Complex Numbers as Roots
- Repeated Roots
Why is this Theorem Important?
See also

What is a Polynomial?

A polynomial is a math expression made of variables (like $x$ ) and numbers, combined using addition, subtraction, and multiplication. The variable can only have whole number powers (like $x^2$ , $x^3$ , but not Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^{0.5} ).

For example, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2x^3 - 5x + 7 is a polynomial.

The numbers $2$ , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -5 , and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 7 are called coefficients.
The variable is $x$ .
The powers of $x$ are $3$ , $1$ (for $x$ ), and $0$ (for the number Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 7 , which is like Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 7x^0 ).

Understanding the Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression.

In Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2x^3 - 5x + 7 , the highest power of $x$ is $3$ . So, its degree is $3$ .
For Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 + 4x - 1 , the highest power is $2$ . Its degree is $2$ .
For Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 5x - 2 , the highest power is $1$ . Its degree is $1$ .

The fundamental theorem of algebra connects the degree of a polynomial to the number of its roots.

What are Polynomial Roots?

A root of a polynomial equation is a value for the variable that makes the whole equation equal to zero. Imagine you have the polynomial Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 - 4 . If you set it equal to zero: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 - 4 = 0 . The roots are the values of $x$ that make this true.

If $x = 2$ , then Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^2 - 4 = 4 - 4 = 0 . So, $2$ is a root.
If Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = -2 , then Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (-2)^2 - 4 = 4 - 4 = 0 . So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -2 is also a root.

For Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 - 4 , the degree is $2$ , and it has exactly $2$ roots ( $2$ and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -2 ). This matches what the theorem says!

Counting All the Roots

The theorem says a polynomial of degree $n$ has exactly $n$ roots. This is true if we consider two special cases:

Complex Numbers as Roots

Sometimes, the roots are not just regular numbers (like Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 1, 2, -5 ). They can be complex numbers. Complex numbers involve the imaginary unit $i$ , where $i^2 = -1$ . For example, the polynomial Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 + 1 has a degree of $2$ . If you set $x^2 + 1 = 0$ , then Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 = -1 . The roots are Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = i and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = -i . These are complex numbers.

Repeated Roots

Sometimes, a root can appear more than once. This is called a "repeated root" or "multiple root." Consider the polynomial Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (x-3)^2 , which is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 - 6x + 9 . Its degree is $2$ . If you set Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (x-3)^2 = 0 , the only value that works is $x = 3$ . But the theorem says there should be two roots. In this case, we count $3$ twice. So, the roots are $3$ and $3$ . This way, the count of roots always matches the degree.

Why is this Theorem Important?

The fundamental theorem of algebra is a cornerstone of mathematics. It tells us that every polynomial equation has a complete set of solutions. This is very useful in many areas of science and engineering, like designing circuits, understanding physics, or even creating computer graphics. It guarantees that we can always find all the answers to these types of equations.

Fundamental theorem of algebra facts for kids

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