Apple and Orange

Combining Like Terms

A polynomial may need to be simplified. One way to simplify a polynomial is to combine the like terms if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[/latex] and [latex]-5x^{2}[/latex] are like terms: They both have [latex]x[/latex] as the variable, and the exponent is [latex]2[/latex] for each. However, [latex]3x^{2}[/latex] and [latex]3x[/latex] are not like terms, because their exponents are different.

Here are some examples of terms that are alike and some that are unlike.

You can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.

[latex]2\left(3+6\right)=2\left(3\right)+2\left(6\right)[/latex]

Both expressions equal [latex]18[/latex]. So you can write the expression in whichever form is the most useful.

Let’s see how we can use this property to combine like terms.

You may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.

When you have a polynomial with more terms, you have to be careful that you combine only like terms. If two terms are not like terms, you can’t combine them.

Adding and Subtracting Monomials

Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

Here is a brief summary of the steps we will follow to add or subtract polynomials.

Pay attention to signs when adding or subtracting monomials.  In the example below, we are subtracting a monomial with a negative coefficient.

Whenever we add monomials in which the variables are not the same, even if their exponents have the same value, they are not like terms and therefore cannot be added together.

Add and Subtract Polynomials

Adding and subtracting polynomials may sound complicated, but it’s really not much different from the addition and subtraction that you do every day.  You can add two (or more) polynomials as you have added algebraic expressions.  Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms—those with the same variables with the same exponent. You can remove the parentheses and then use the Commutative Property to rearrange the terms to put like terms together. (It may also be helpful to underline, circle, or box like terms.)

When you are adding polynomials that have subtraction, it is important to remember to keep the sign on each term as you are collecting like terms.  The next example will show you how to regroup terms that are subtracted when you are collecting like terms.

Example

Find the sum: [latex]\left(4{x}^{2}-5x+1\right)+\left(3{x}^{2}-8x - 9\right)[/latex].

Show Solution

Solution

	[latex]\left(4{x}^{2}-5x+1\right)+\left(3{x}^{2}-8x - 9\right)[/latex]
Identify like terms.
Rearrange to get the like terms together.
Combine like terms.	[latex]7x^2-13x-8[/latex]

The above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this “vertical” method of adding polynomials:

Sometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn’t so tidy. When there isn’t a matching like term for every term, there will be empty places in the vertical arrangement.

You may be thinking, how is this different than combining like terms, which we did in the last section? The answer is, it’s not really. We just added a layer to combining like terms by adding more terms to combine. :) Polynomials are a useful tool for describing the behavior of anything that isn’t linear, and sometimes you may need to add them.

In the following video, you will see more examples of combining like terms by adding polynomials.

In the next section we will show how to subtract polynomials.

Find the opposite of a polynomial

When you are solving equations, it may come up that you need to subtract polynomials. This means subtracting each term of a polynomial, which requires changing the sign of each term in a polynomial. Recall that changing the sign of [latex]3[/latex] gives [latex]−3[/latex], and changing the sign of [latex]−3[/latex] gives [latex]3[/latex]. Just as changing the sign of a number is found by multiplying the number by [latex]−1[/latex], we can change the sign of a polynomial by multiplying it by [latex]−1[/latex]. Think of this in the same way as you would the distributive property.  You are distributing [latex]−1[/latex] to each term in the polynomial.  Changing the sign of a polynomial is also called finding the opposite.

Be careful when there are negative terms or subtractions in the polynomial already.  Just remember that you are changing the sign, so if it is negative, it will become positive.

Notice that in finding the opposite of a polynomial, you change the sign of each term in the polynomial, then rewrite the polynomial with the new signs on each term.

Subtract polynomials

When you subtract one polynomial from another, you will first find the opposite of the polynomial being subtracted, then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.

When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.

In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.

When you have many terms, like in the examples above, try the vertical approach shown above to keep your terms organized.  However you choose to combine polynomials is up to you—the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.

When we add polynomials as we did in the last example, we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.

example

Find the difference: [latex]\left(7{u}^{2}-5u+3\right)-\left(4{u}^{2}-2\right)[/latex].

Show Solution

Solution

	[latex]\left(7{u}^{2}-5u+3\right)-\left(4{u}^{2}-2\right)[/latex]
Distribute and identify like terms.
Rearrange the terms.
Combine like terms.	[latex]3{u}^{2}-5u+5[/latex]

Exercises

Subtract [latex]\left({m}^{2}-3m+8\right)[/latex] from [latex]\left(9{m}^{2}-7m+4\right)[/latex].

Show Solution

Solution

Subtract [latex]\left({m}^{2}-3m+8\right)[/latex] from [latex]\left(9{m}^{2}-7m+4\right)[/latex].	[latex](9{m}^{2}-7m+4)-({m}^{2}-3m+8)[/latex]
Distribute and identify like terms.
Rearrange the terms.
Combine like terms.	[latex]8m^2-4m-4[/latex]

In the following video, you will see more examples of subtracting polynomials.

In the next video we show more examples of adding and subtracting polynomials.

Summary

We have seen that subtracting a polynomial means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are alike.  How you organize this process is up to you, but we have shown two ways here.  One method is to place the terms next to each other horizontally, putting like terms next to each other to make combining them easier.  The other method was to place the polynomial being subtracted underneath the other after changing the signs of each term. In this method it is important to align like terms and use a blank space when there is no like term.

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