A monomial is an algebraic expression where the only operations between the variables are the product and power of the natural exponent.
Elements of a Monomial
The coefficient of a monomial is the number that multiplies the variable(s).
The literal part is formed by the variables (letters) and its exponents.
The degree of a monomial is the sum of all exponents of the letters or variables.
The degree of 2x2 y3 z is: 2 + 3 + 1 = 6
Two monomials are similar when they have the same literal part.
2x2y3z is similar to 5x2y3z
In order to add monomials, they have to be similar.
The sum of the monomials is another monomial whose literal part is the same and the coefficient is the sum of the coefficients.
axn + bxn = (a + b)xn
2x2 y3 z + 3x2 y3 z = 5x2 y3 z
The sum of two or more non-similar monomials is a polynomial.
2x2 y3 + 3x2 y3 z
The product of two or more monomials is another monomial whose coefficient is the product of the coefficients and whose literal part is product of the powers with the same base.
axn · bxm = (a · b)(xn · x m) = (a · b)xn + m
(5x2 y3 z) · (2 y2 z2) = 10 x2 y5 z3
Monomials can only be divided if they have the same literal part and the degree of the dividend is greater than or equal to the corresponding divisor.
The division of monomials is another monomial whose coefficient is the quotient of the coefficients and the literal part is the quotient of the powers with the same base.
axn : bxm = (a : b) (xn : x m) = (a : b)xn − m
If the degree of the divisor is greater, an algebraic fraction is obtained.
Power of a Monomial
To calculate the power of a monomial, every element of the monomial is raised to the exponent of the power.
(axn)m = am · xn · m
(2x3)3 = 23 · (x3)3 = 8x9
(−3x2)3 = (−3)3 · (x3)2 = −27x6