Operations with Monomials
Addition of Monomials
Similar monomials can be added.
axn + bxn = (a + b)xn
2x2y3z + 3x y3z = 5x2y3z
If the monomials are not similar, a polynomial is obtained.
2x2y3 + 3x2y3 z
Multiplication of a Number by a Monomial
The product of a number by a monomial is another similar monomial whose coefficient is the product of the coefficient of the monomial and the number.
5 · (2x2 y3 z) = 10x2 y3 z
Multiplication of Monomials
The multiplication of monomials is another monomial that takes as its coefficient the product of the coefficients and whose literal part is obtained by multiplying the powers that have the same base.
axn · bxm = (a · b)(xn · x m) = (a · b)xn + m
(5x2 y3 z) · (2 y2 z2) = 10 x2 y5 z3
Division of Monomials
Dividing monomials can only be performed if they have the same literal part and the degree of the dividend has to be greater than or equal to the corresponding divisor.
The division of monomials is another monomial whose coefficient is the quotient of the coefficients and its literal part is obtained by dividing the powers that have the same base.
axn : bxm = (a : b) (xn : x m) = (a : b)xn − m
If the degree of divisor is greater, an algebraic fraction is obtained.
Power of a Monomial
To determine the power of a monomial, every element in 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 · (x2)3 = −27x6