How are the processes of adding monomials and adding polynomials alike? How are the processes different?
The processes of adding monomials and adding polynomials are alike in the sense that both involve combining terms with similar variables.
To add monomials, you simply add the coefficients of the monomials if they have the same variable(s) raised to the same power(s). For example, if you want to add 2x and 3x, the result is 5x. Similarly, if you want to add 5x^2 and 2x^2, the result is 7x^2.
Adding polynomials follows a similar principle. A polynomial is an expression that consists of multiple terms, each of which can be a monomial. To add polynomials, you combine like terms by adding their coefficients. Like terms are those with the same variables raised to the same power(s). For example, to add 3x^2 + 4x - 2x^2 - 5x, you first combine the like terms (3x^2 and -2x^2) to get x^2, then combine the like terms with x (4x and -5x) to get -x. The final result is x^2 - x.
However, the processes of adding monomials and adding polynomials also have differences. The main difference is that when adding polynomials, you need to ensure that you are adding the corresponding terms with the same variables. In other words, you may need to add zero coefficients for variables that are missing in some terms.
For example, if you want to add 3x^3 + 2x^2 + 5x - 4 and 2x^3 + x^2 - 3x + 2, you start by arranging the terms in descending order based on the exponent of x:
(3x^3 + 2x^2 + 5x - 4)
+ (2x^3 + x^2 - 3x + 2)
Now, you add the corresponding terms:
(3x^3 + 2x^2 + 5x - 4)
+ (2x^3 + x^2 - 3x + 2)
-----------------------
5x^3 + 3x^2 + 2x - 2
Notice that in this example, there are no terms with x^1 (x without an exponent) in the final result, but we still need to include it with a coefficient of 0 to indicate the absence of that term.
In summary, the processes of adding monomials and adding polynomials are alike in terms of combining like terms by adding their coefficients. However, adding polynomials also involves adding terms with missing variables by including them with a coefficient of 0.