Monic polynomial

On this post you will find what a monic polynomial is and examples of monic polynomials. You will also see the properties of this type of polynomial and how a polynomial is transformed into a monic polynomial.

What is a monic polynomial?

The definition of a monic polynomial is as follows:

In mathematics, a monic polynomial is a univariate polynomial (polynomial with only one variable) whose leading coefficient is equal to 1.

For example, the following polynomial of degree 2 is monic because it is a single-variable polynomial and its leading coefficient is 1:

x^2+3x-5

Remember that the leading coefficient of a polynomial is the coefficient of its highest degree term.

Examples of monic polynomials

Once we have seen what a monic polynomial means, let’s see several examples of this type of polynomial:

Example of monic polynomial of degree 2:

x^2-6x-4

Example of monic polynomial of degree 3:

x^3+4x^2+x+10

Example of monic polynomial of degree 4:

x^4+5x+6

How to transform any polynomial into a monic polynomial

Now that we know the meaning of a monic polynomial, let’s see how to convert a polynomial into a monic polynomial.

So, we are going to solve a problem step by step to see how to do it:

P(x)=4x^5+3x^4-8x^2+2x-12

To transform the polynomial into a monic one, we have to divide all the terms of the polynomial by the coefficient of the highest degree term. In this case, the coefficient of the highest degree term is 4, therefore:

 \begin{aligned} \cfrac{P(x)}{4} & =\cfrac{4x^5}{4}+\cfrac{3x^4}{4}-\cfrac{8x^2}{4}+\cfrac{2x}{4}-\cfrac{12}{4} \\[2ex] & = \cfrac{4}{4}x^5+\cfrac{3}{4}x^4-\cfrac{8}{4}x^2+\cfrac{2}{4}x-\cfrac{12}{4} \end{aligned}

Now we simplify the fractions of the polynomial:

1x^5+\cfrac{3}{4}x^4-2x^2+\cfrac{1}{2}x-3

x^5+\cfrac{3}{4}x^4-2x^2+\cfrac{1}{2}x-3

And we have already converted the polynomial of the problem into a monic polynomial.

Properties of monic polynomials

Monic polynomials meet the following characteristics:

  • The product of a monic polynomial and another monic polynomial always results in a monic polynomial.

This is due to the properties of the polynomial multiplication. To know why this always happens, see the properties of the product of polynomials.

  • If a monic polynomial is composed only of integer coefficients, the roots of that monic polynomial will also be integers.

The roots (or zeros) of a polynomial are numbers that define a polynomial, therefore, it is a very important concept. If you do not know what they are or how to calculate them, you should see:

How to find the roots of a polynomial

  • Even if the leading coefficient of a multivariate polynomial is one, this type of polynomial is never considered a monic polynomial because it has more than one variable.

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