## How to add and subtract polynomials

When adding and subtracting polynomials, you’ll have to deal with combining like terms and also be aware of the order of operations within the question. A point you’ll have to take note of before we begin so that you don’t make any mistakes is to be careful with minus signs.

What are like terms? Like terms are terms whose variables are the same. An example will be that 3x2 and 11x2 are like terms since their variables are both x2. However, 3x2 and 6x are not like terms, because one variable is x-squared whereas one is just x. You can see, however, that the coefficients do not have to be the same. In the first example just now, 3 and 11 are not the same, but they can still be combined because their variables are identical.

Let’s take a look at this example, which can demonstrate how adding and subtracting polynomials work. We’ll be carrying out basic operations with polynomials.

Question:

$\left( {x - 4xy - 2y} \right) + \left( {3xy - y} \right) + \left( { - 6x - 5y} \right)$

Solution:

$\left( {x - 4xy - 2y} \right) + \left( {3xy - y} \right) + \left( { - 6x - 5y} \right)$

1. Take out the parentheses

$x - 4xy - 2y + 3xy - y - 6x - 5y$

2. Look for like terms

$x - 4xy - 2y + 3xy - y - 6x - 5y$

$- 5x - 1xy - 7y$

We’ve outlined the three basic steps to solving a problem that deals with parentheses as well as both addition and subtraction. Let’s look more in depth into each of the steps.

In the first step, we’re removing the parentheses. This helps us identify the polynomials that we’ll have to work with. Remember our note about paying attention to plus or minus signs? This is going to come in very handy soon. Since the signs outside the parentheses are both “+”, it makes removing the parentheses a lot easier.

Now that you’ve got all your terms, it’s time to find the ones that are “like terms”. We’ve got a variety of variables, including “x”, “xy”, and “y”. In step number 2, you can see how we’ve put all the terms that have the same variables together to get ready to add polynomials or subtract polynomials. Always take the sign in front of your term with you when you move them around. Otherwise, you’ll get the wrong answer, and may accidentally subtract when you’re supposed to add, or vice versa.

Working from left to right, start by adding polynomials, then subtracting polynomials in order. If there were powers in the variables, you would usually show your answer in order of descending powers. This means you may have to reorder your terms for your final answer.

In this example, we learned how to add and subtract polynomials horizontally. But similar to regular adding and subtracting, you can also do it vertically. For both methods, you’ll end up with the same answer, so it’s mostly up to you whether you prefer to do it vertically or horizontally. You may find that for simple additions, using the horizontal method is easier since you won’t have to rewrite the problem. However, as you progress into harder questions, the vertical method can help you ensure you don’t forget terms or minus signs.

For more examples, here’s an interactive one that can give you through steps of polynomial addition/subtraction questions you type in. For a more in depth look at like terms, this is a link that will help.

##### Don't just watch, practice makes perfect.

Adding polynomials is basically combining the like terms together. Like terms are the terms with the same variables and degree. Subtracting polynomials is very similar to that, but you will need to reverse the sign of each term to get rid of the like terms.

#### Lessons

• 1.
a)
$\left( {4 - 2x + 3{x^2}} \right) + \left( { - x - 4{x^2} + 7} \right)$

b)
$\left( {7a + 1} \right) + \left( { - 4 - 3a} \right)$

c)
$\left( {{n^2} - 5} \right) + \left( {{n^2} + 6} \right)$

d)
$\left( x - 4xy - 2y \right) + \left( {3xy - y} \right) + \left( { - 6x - 5y} \right)$

• 2.
Write the opposite of each expression.
a)
$- 2{x^3} + 5x - 4.6$

b)
$7n - 3$

c)
${y^2} - 8y + 1$

• 3.
Subtract the following polynomials.
a)
$\left( { - 2{x^2} - 6x + 3} \right) - \left( {3{x^2} - x - 8} \right)$

b)
$\left( { - {x^2} + 7 - 3x} \right) - \left( {8 - x} \right)$

c)
$\left( {5{x^2} - 3x} \right) - \left( {2x - {x^2}} \right)$

d)
$\left( { - xy + 3x - 3} \right) - \left( { - x - 5 + 6xy} \right)$

• 4.
Combine like terms.
a)
$\left( {{a^2} - 3a} \right) + \left( {2{a^2} - 4} \right) - \left( {6a - 1} \right)$

b)
$\left( {x + 5} \right) + \left( {2x - 3} \right) + \left( {7x - 6} \right)$

c)
$\left( { - 4x - 3.1} \right) - \left( { - 5.6x - 2} \right) - \left( {1.1x - 0.6} \right)$

d)
$\left( {3y - 2} \right) + \left( {y - 6} \right) + \left( {10y - 5} \right)$