Figure 1 shows a monic polynomial with four terms. 

Figure 1 – Demonstration of a Monic Polynomial

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.

Figure 2 – Demonstration of Two Non-monic 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.

x² – 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:

x² + 6x + 8 = 0

To factor this equation, it can be written as:

x² + 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:

px² + qx + r = 0

Where:

p ≠ 0

Where p, q, and r are the coefficients of x², x, and $x⁰$. For a quadratic equation to be monic, p should be equal to 1.

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

Figure 3 – Examples of Monic Binomial and Monic 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 x³ + 2x² + 4x + 1 is a monic polynomial with degree 3.

Monic Polynomial With Degree Four

The polynomial 5t² + t⁴ + 5 is a variable “t” monic polynomial with degree 4.

Monic Polynomial With Degree Five

The polynomial s⁵ + 2s⁴ + 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) = 5n⁴ + 35n³ + 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) = n⁴ + 7n³ + 2n

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

Figure 4 – 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) = m⁴ + 2m³ + 3 and Q(m) = m² + 7 is:

R(m).Q(m) = (m⁴ + 2m³ – 3)(m² + 7)

P(m) = m⁶ + 7m⁴ + 2m⁵ + 14m³ – 3m² – 21

P(m) = m⁶ + 2m⁵ + 7m⁴ + 14m³ – 3m² – 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 x² + 8x + 16 is solved by using the formula:

(a + b)² = a² + 2ab + b²

(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 x² + 8x + 16 with integer coefficients are also integers.

Example

Which of the following is a monic polynomial?

(a) x² + 2y – 3

(b) y² + 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:

y² + y – 42 = 0

It can be solved by factorization as:

y² + 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 y² + y – 42 are 6 and -7.

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