Monic Polynomial

A monic polynomial is a polynomial in which the coefficient of the term with the highest degree (the leading term) is 1. 

Mathematically, for a polynomial in general form P(x) = a_nxⁿ + a_{n – 1}x^{n – 1} + … + a₁x + a₀, it is monic if a_n is 1

For example,

In the polynomial P(x) = x³ – 4x + 7, the leading coefficient is 1. Thus, P(x) is a monic polynomial.

Here are a few more examples of monic polynomials:

• Q(x) = x² + 5x + 6 (monic quadratic polynomial)
• R(x) = x³ – 25 (monic cubic polynomial)
• T(x) = x⁴ + 2x² + 1 (monic quartic polynomial)
• F(x) = x – 11 (monic linear polynomial)

Note: 1 is a monic polynomial, as the leading coefficient of the polynomial is 1.

However, the following polynomials are not monic as their leading coefficients are not 1:

• f(x) = 7x² + 6
• g(x) = 3x⁴ – 2x² + x – 4
• k(x) = 2x⁷ – 11x⁵ + 5

In summary, a monic polynomial must have a leading term where the highest power of the variable is not multiplied by anything other than 1.

Solved Examples

Determine whether the following polynomials are monic:
a) x³ + 2x² – x + 4
b) 5x² – 3x + 1
c) x⁴ – x³ + x² – x + 1

Solution:

a) Given, x³ + 2x² – x + 4
Here, the leading coefficient is 1
Thus, it is a monic polynomial.
b) Given, 5x² – 3x + 1
Here, the leading coefficient is 5
Thus, it is not a monic polynomial.
c) Given, x⁴ – x³ + x² – x + 1
Here, the leading coefficient is 1
Thus, it is a monic polynomial.

Convert the following polynomials to monic form: 
a) 4x³ – 8x + 12
b) 2x² + 6x + 10

Solution:

a) Given, f(x) = 4x³ – 8x + 12
On dividing each term by 4,
g(x) = x³ – 2x + 3
Here, the leading coefficient is 1
Thus, g(x) is a monic polynomial.
b) Given, f(x) = 2x² + 6x + 10
On dividing each term by 2,
g(x) = x² + 3x + 5
Here, the leading coefficient is 1
Thus, g(x) is a monic polynomial.