Combining Like Terms
When adding and subtracting polynomials, it really is an exercise in combining like terms. Like terms have the same variables, which are raised to the same power. For example, 2x2 and 7x2 are like terms, because they both contain x raised to the second power. However, 2x3 and 7x2 are not like terms. Although they both contain the variable x, their exponents are different: one is x cubed, and the other is x squared.
Like terms can be combined by adding their coefficients. For example, combining 2x2 and 7x2 gives us 9x2. If coefficients are negative, we can think of it as subtraction: combining 2x2 and –7x2 gives us –5x2
Polynomials have several terms, and likely contain several different types of terms. When we add and subtract polynomials, it is important to identify like terms, and organize the terms by their type, so that we can easily add or subtract their coefficients.
Identifying Like Terms
Consider this group of terms:
Let's examine the variables and exponents to determine which terms are like terms and can be combined. When looking at the term 2xy, we scan the other terms to see if we have other terms with an x and a y. We see 3x2y, but is this a like term to 2xy? It actually isn't. This is because 3x2y has two factors of x (the exponent of 2 is attached to the x), whereas 2xy only has 1 factor of x.
We do however have the like terms –x2 and 4x2. These can be combined to simplify to 3x2, because –1 and 4 combine to 3.
If the terms above all belonged to the same polynomial, we could write the polynomial as:
It is standard to write the terms in order of their degree, from highest to lowest. 3x2y is written first because its degree is 3, whereas the other terms have a degree of 2.
When adding (and subtracting) polynomials, it helps to write the problem vertically, so that you can align the like-terms, making addition (or subtraction) of the coefficients easier to do in your head.
Let's add 3x3 + 2x – 5 to 2x3 + 5x2 – 7x:
Before we complete the addition, let's take a closer look at the vertical set up. The whole point of writing the addition vertically is to line up the like terms. But we noticed that there were some terms in one polynomial that weren't in the other. In this case, it helps to write a term with a coefficient of zero, as a placeholder to keep everything vertically aligned.
When subtracting polynomials, we follow the same procedure, only we subtract coefficients rather than add them. The only tricky thing to watch out for is subtracting a negative number. This should be thought of as adding a positive number.
Let's subtract 2x2 – 7x – 3 from 6x3 – 4:
Once again, we write the problem vertically, and add terms with coefficients of 0 to keep everything aligned. It also helps to group each polynomial in parentheses, so that we can get the subtraction out in front and still be aware of the signs of each term within the polynomial.
Was any part of that subtraction tricky? It was probably easy enough to get the coefficients of the first two terms: 6 – 0 = 6, and 0 – 2 = –2. However, sometimes it can be difficult to remember signs with subtraction, especially for the last two coefficients: 0 – (–7) = 0 + 7 = 7, and –4 – (–3) = –4 + 3 = –1