We want to compare the processes of adding monomials and adding polynomials. We will begin by recalling the definitions of a monomial and a polynomial.
A monomial can be real number, a variable, a product of a real number and one or more variables. The exponents of the variables must be whole numbers.
3, sqrt(7), 5x^2, 2ab
A polynomial is an algebraic expression that can be a monomial or a sum of monomials.
5x^2, x^2+x, x^3+2x+3
Now, let's examine the processes of adding monomials.
Adding Monomials
Note that to perform addition in monomials, they must be like terms.To add monomials, we add the coefficients of like monomials and keep the variable(s) and their exponents unchanged.
Monomial I
Monomial II
Sum
3x
4x
( 3+ 4)x=7x
7
11
7+ 11=18
-2x^2
2x^2
( -2+ 2)x^2=
As we can see, if the monomials do not cancel each other out, the sum has only one term. In this case, the degree of the sum is the same as the degree of the monomials. If the monomials cancel each other out, the sum has no degree.
If the polynomials cancel each other out, the sum has no degree. However, if they do not cancel each other, the degree is less than or equal to the degree of the polynomial that has the greatest degree.
Comparison
Finally, we can compare the process of adding monomials and adding polynomials considering the above information.
Similarity:
In both additions, we add the coefficients of like terms.
If the terms cancel each other, the sum has no degree.
Differences:
If the terms do not cancel each other out, the of two monomials has only one term, while the sum of polynomials has at least one term.
If the terms do not cancel each other, the degree of the sum of monomials is the same as the degree of the monomials added. For the polynomials, the degree is less than or equal to the degree of the polynomial with the greatest degree.
