Addition and Subtraction Activities

When you add the terms that a polynomial contains, one of the first steps is to group like terms. Like terms identify what you are adding. Here is an equation with like terms:

2x + 3x

In this instance, the terms of the equation constitute like terms because each term consists of the variable x. Each of the like terms is associated with an integer coefficient. To solve for x, you put the distributive property to work:

(2 + 3)x = 5x

Here is an extended example of the same type of operation:

6x + 7x − 4x + 8x − x

= (6x + 7x) − 4x + 8x − x

= (13x − 4x)+ 8x − x

= (9x + 8x) − x

= (17x − x)

= 16x

For each step, you successively group like terms, and then carry out additions or subtractions. In the end, you have combined all terms in the polynomial and are left with the final term.

You can also group terms with exponents. Here are a couple of expressions that you can more easily view as monomials if you combine the like terms they contain:

x2·x5 = x2 + 5 = x7	Add the exponents with the same base value.
x3·x−2 = x3−2 = x1 = x

If you consider that the commutative and associate properties of numbers allow you to express exponents in different ways, you can extend the work you perform by grouping like terms. Consider this expression:

6x²y·8xy³

6·8·x²x¹·y¹y³ Group like terms.

48x³y⁴

Here is an example that incorporates negative exponents. While the grouping creates a fraction, it still serves to simplify the terms:

Using the distributive property, you can regroup like terms along the following lines:

6x1y2(3x4 − 5xy2)
= 6x1y2(3x4)− 6x1y2(5x1y2)	Distribute the multiplications.
= 18x5y2 − 30x2y4

With this expression, you begin by using the distributive property to extract the terms beginning with the coefficients 3 and 5 from the parentheses. You can then group the resulting line terms and multiply the variables. To perform the multiplications, you add the exponents.