Adding and subtracting polynomials may sound complicated, but it’s really not much different from the addition and subtraction that you do every day. The main thing to remember is to look for and combine like terms. You can add two (or more) polynomials as you have added algebraic expressions. You can remove the parentheses and combine like terms.
Example 1
Add the following two binomials. [latex]\left(3b+5\right)+\left(2b+4\right)[/latex]
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.
Collect like terms, making sure you keep the sign on each term. For example, when you collect the [latex]x^2[/latex] terms, make sure to keep the negative sign on [latex]-5x^2[/latex].
Helpful Hint: We find that it is easier to put the terms with a negative sign on the right of the terms that are positive. This would mean that the [latex]x^2[/latex] terms would be grouped as [latex]\left(3x^{2}-5x^{2}\right)[/latex]. If both terms are negative, then it doesn’t matter which is on the left or right.
The polynomial now looks like this, with like terms collected:
The [latex]x^2[/latex] terms will simplify to [latex]-2x^{2}[/latex]
The [latex]x[/latex] will simplify to [latex]-3x[/latex]
The constant terms will simplify to [latex]-2[/latex]
Rewrite the polynomial with it’s simplified terms, keeping the sign on each term.
[latex]-2x^{2}-3x-2[/latex]
As a matter of convention, we write polynomials in descending order based on degree. Notice how we put the [latex]x^2[/latex] term first, the [latex]x[/latex] term second and the constant term last.
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.
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 3 gives [latex]−3[/latex], and changing the sign of [latex]−3[/latex] gives 3. 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.
Example 5
Find the opposite of [latex]9x^{2}+10x+5[/latex].
Show Solution
Find the opposite by multiplying by [latex]−1[/latex].
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.
Example 6
Find the opposite of [latex]3p^{2}–5p+7[/latex].
Show Solution
Find the opposite by multiplying by [latex]-1[/latex].
Now you can rewrite the polynomial with the new sign on each term:
[latex]-3p^{2}+5p-7[/latex]
Answer
The opposite of [latex]3p^{2}-5p+7[/latex] is [latex]-3p^{2}+5p-7[/latex].
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.
Subtracting 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.
When you have many terms, like in the example above, try the vertical approach from the previous page 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.
In the following video, you will see more examples of 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.
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 3 gives [latex]−3[/latex], and changing the sign of [latex]−3[/latex] gives 3. 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.