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# Monic Polynomial|Definition & Meaning

## Definition

A **monic** polynomial is a **one-variable** polynomial with the coefficient of the highest **power** variable equal to **one**. In other words, it is a **univariate** polynomial with a leading **coefficient** as one.

**Figure 1** shows a **monic** polynomial with **four** terms.

The **degree** of the above polynomial is** four**. Notice that it is only a **one-variable** “**x**” polynomial with the variable’s **coefficient** of the highest **power** “four” equal to** one**, making it** monic**.

Both **conditions** must be **fulfilled** for a polynomial to be monic. **Figure 2** shows two polynomials.

The **first** polynomial is not monic because it is a **two-variable** polynomial of **a** and **b**. The **second** polynomial is a one-variable polynomial of variable **m** but it is also not monic as the **coefficient** of the highest power variable is **three** not one.

## Types of Monic Polynomial

A **polynomial** is an **expression** consisting of **variables**, constants, and exponents combined using math **operators**. The word “**poly**” means “**many**”, hence a polynomial has one or more than one term.

The **types** of **monic** polynomials include the monic **binomial** and the **trinomial**.

### Monic Binomial

A **binomial** is a polynomial consisting of **two** terms. A **monic** binomial is a **one-variable** polynomial having two **terms** with the leading **coefficient** as one. For example, the following **equation** consists of a monic binomial.

x^{2} – 3x = 0

The **degree** of this monic binomial is** 2**. This **equation** can be **solved** by taking **x** as common:

x(x – 3) = 0

The above equation can also be written as:

x = 0, x – 3 = 0

x = 0, x = 3

So, the **roots** of this monic binomial are **x = 0** and **x = 3**. The **number** of roots of a polynomial depends upon the **degree** of the polynomial.

### Monic Trinomial

A monic **trinomial** is a **univariate** polynomial with **three** terms having the variable’s coefficient of the **highest** degree equal to one. The following equation has a **monic** trinomial:

x^{2} + 6x + 8 = 0

To **factor** this equation, it can be written as:

x^{2} + 4x + 2x + 8 = 0

x(x + 4) + 2(x + 4) = 0

Taking **(x + 4)** as common gives:

(x + 4)(x + 2) = 0

This can be written as:

(x + 4) = 0, (x + 2) = 0

So:

x = -4, x = -2

So, the **roots** of this monic **trinomial** equation are **-4** and **-2**.

A **quadratic** polynomial is a **second-degree** polynomial consisting of **three** terms hence it is also a **trinomial**. The quadratic equation is:

px^{2} + qx + r = 0

Where:

p ≠ 0

Where **p**, **q**, and **r** are the **coefficients** of x^{2}, x, and $x^{0}$. For a **quadratic** equation to be **monic**, **p** should be equal to **1**.

**Figure 3** shows examples of a **monic** binomial and a trinomial.

## Monic Polynomials With Different Degrees

The **degree** of a polynomial is the **highest** power of the **variable** in the polynomial. **Identifying** the degree of a **monic** polynomial is easier as the variable with the highest degree will have a **coefficient** of one. Some monic **polynomials** with different degrees are discussed below:

### Monic Polynomial With Degree Three

The polynomial x^{3} + 2x^{2} + 4x + 1 is a **monic** polynomial with degree **3**.

### Monic Polynomial With Degree Four

The **polynomial** 5t^{2} + t^{4} + 5 is a variable “**t**” monic polynomial with degree **4**.

### Monic Polynomial With Degree Five

The **polynomial** s^{5} + 2s^{4} + 7s + 6 is a **monic** polynomial with variable** s** having degree **5**.

## Conversion of a Non-Monic Polynomial Into a Monic Polynomial

A **non-monic** polynomial can be converted into a **monic** polynomial by **dividing** the whole polynomial by the **integer** multiplied by the highest **power** variable. For this** conversion**, the polynomial must be a single **variable** polynomial.

For **example**, a non-monic **polynomial** is given as:

P(n) = 5n^{4} + 35n^{3} + 10n

It is a **degree 4** non-monic trinomial with a **single** variable **n**. It can be converted to a **monic** polynomial by **dividing** it by **5**, so the polynomial becomes:

P(n) = n^{4} + 7n^{3} + 2n

**Figure 4** shows the **conversion** of a non-monic polynomial into a monic polynomial.

## Properties of Monic Polynomial

A **monic** polynomial has the following important **characteristics**.

### Product of Monic Polynomials

The product of **two** monic polynomials is also a **monic** polynomial provided that both have the **same** variable. For example, the product **P(m)** of **R(m)** = m^{4} + 2m^{3} + 3 and **Q(m)** = m^{2} + 7 is:

R(m).Q(m) = (m^{4} + 2m^{3} – 3)(m^{2} + 7)

P(m) = m^{6} + 7m^{4} + 2m^{5} + 14m^{3} – 3m^{2} – 21

P(m) = m^{6} + 2m^{5} + 7m^{4} + 14m^{3} – 3m^{2} – 21

Hence, **P(m)** is also a **monic** polynomial of degree** 6**.

### Roots of a Monic Polynomial

The **roots** of a polynomial are the solutions or the **values** of the variables for which the **polynomial** is equal to **zero**. If a monic polynomial has all the **coefficients** as integers, then its roots will also be **integers**.

For example, the **monic** polynomial x^{2} + 8x + 16 is solved by using the **formula**:

(a + b)^{2} = a^{2} + 2ab + b^{2}

(x)^{2} + 2(x)(4) + (4)^{2} = 0

(x + 4)^{2} = 0

It can be written as:

(x + 4) = 0, (x + 4) = 0

x = -4, x = -4

Hence, the **roots** of the polynomial x^{2} + 8x + 16 with integer **coefficients** are also **integers**.

## Example

Which of the following is a **monic** polynomial?

(a) x^{2} + 2y – 3

(b) y^{2} + y – 42

Is the monic polynomial **binomial** or a** trinomial**? What is its **degree**? Also, find its **roots**.

### Solution

The polynomial (**a**) is a **two-variable** polynomial hence it is **not** a **monic** polynomial.

Polynomial (**b**) is a **monic** polynomial. It is a **trinomial** with a degree of **2**. The **roots** of the polynomial can be found by equating it to zero as:

y^{2} + y – 42 = 0

It can be solved by **factorization** as:

y^{2} + 7y – 6y – 42 = 0

y(y + 7) – 6(y + 7) = 0

Taking **(y + 7)** as common:

(y + 7)(y – 6) = 0

So:

(y + 7) = 0, (y – 6) = 0

y = -7, y = 6

So, the **roots** of the **monic** polynomial y^{2} + y – 42 are **6** and **-7**.

*All the images are created using GeoGebra.*