Learning Outcomes
- Combine like terms through addition and subtraction
- Add and subtract monomials
- Add and subtract polynomials
Key words
- Like terms: Terms that have identical variables with identical exponents
- Sum: the answer when two or more terms are added
- Difference: the answer when two or more terms are subtracted
- Commutative property of addition: reordering terms in an expression results in an equivalent expression [latex]a+b=b+a[/latex]
- Associative property of addition: regrouping terms in an expression results in an equivalent expression [latex](a+b)+c=a+(b+c)[/latex]
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 identical variables with the identical exponents. 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 like terms and unlike terms.
| Term | Like Terms | UNLike Terms |
| [latex]a[/latex] | [latex]3a, \,\,\,-2a,\,\,\, \frac{1}{2}a[/latex] | [latex]a^2,\,\,\,\frac{1}{a},\,\,\, \sqrt{a}[/latex] |
| [latex]a^2[/latex] | [latex]-5a^2,\,\,\,\frac{1}{4}a^2,\,\,\, 0.56a^2[/latex] | [latex]\frac{1}{a^2},\,\,\,\sqrt{a^2},\,\,\, a^3[/latex] |
| [latex]ab[/latex] | [latex]7ab,\,\,\,0.23ab,\,\,\,\frac{2}{3}ab,\,\,\,-ab[/latex] | [latex]a^2b,\,\,\,\frac{1}{ab},\,\,\,\sqrt{ab} [/latex] |
| [latex]ab^2[/latex] | [latex]4ab^2,\,\,\, \frac{ab^2}{7},\,\,\,0.4ab^2,\,\,\, -a^2b[/latex] | [latex]a^2b,\,\,\, ab,\,\,\,\sqrt{ab^2},\,\,\,\frac{1}{ab^2}[/latex] |
Example
Which of these terms are like terms?
[latex]7x^{3}, 7x, 7y, -8x^{3}, 9y, -3x^{2}, 8y^{2}[/latex]
Solution
Like terms must have the same variables, so first identify which terms use the same variables.
[latex]\begin{array}{l}x:7x^{3}, 7x, -8x^{3}, -3x^{2}\\y:7y, 9y, 8y^{2}\end{array}[/latex]
Like terms must also have the same exponents. Identify which terms with the same variables also use the same exponents.
The [latex]x[/latex]–terms [latex]7x^{3}[/latex] and [latex]-8x^{3}[/latex] have the same exponent.
The [latex]y[/latex]–terms [latex]7y[/latex] and [latex]9y[/latex] have the same exponent.
Answer
[latex]7x^{3}[/latex] and [latex]-8x^{3}[/latex] are like terms.
[latex]7y[/latex] and [latex]9y[/latex] are like terms.
We can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition over multiplication 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 we can write the expression in whichever form is the most useful.
Let’s see how we can use this property to combine like terms.
Example
Simplify [latex]3x^{2}-5x^{2}[/latex].
Solution
[latex]3x^{2}[/latex] and [latex]5x^{2}[/latex] are like terms.
[latex]3\left(x^{2}\right)-5\left(x^{2}\right)[/latex]
Rewrite the expression as the product of the difference:
[latex]\left(3-5\right)\left(x^{2}\right)[/latex]
Calculate [latex]3–5[/latex]:
[latex]\left(-2\right)\left(x^{2}\right)[/latex]
Write the sum as the new coefficient:
[latex]-2x^2[/latex]
Answer
[latex]3x^{2}-5x^{2}=-2x^{2}[/latex]
Notice that combining like terms through addition or subtraction involves adding or subtracting the coefficients to find the new coefficient of the like term.
Example
Simplify [latex]6a^{4}+4a^{4}[/latex].
Solution
Notice that both terms have a number multiplied by [latex]a^{4}[/latex]. This makes them like terms.
[latex]6a^{4}+4a^{4}[/latex]
Combine the coefficients, [latex]6[/latex] and [latex]4[/latex].
[latex]\left(6+4\right)\left(a^{4}\right)[/latex]
Calculate the sum.
[latex]\left(10\right)\left(a^{4}\right)[/latex]
Write the sum as the new coefficient.
[latex]10a^4[/latex]
Answer
[latex]6a^{4}+4a^{4}=10a^{4}[/latex]
When we have a polynomial with more terms, we have to be careful that we combine only like terms. If two terms are not like terms, we can’t combine them.
Example
Simplify [latex]3x^{2}-3x+x+1+5x[/latex]
Solution
First identify which terms are like terms: only [latex]-3x[/latex], [latex]x[/latex], and [latex]5x[/latex] are like terms.
Write the polynomial in standard (descending) form and group the like terms:
[latex]\begin{array}{l}3x^{2}-3x+x+1+5x\\3x^{2}+\left(3x+x+5x\right)+1\end{array}[/latex]
Add the coefficients of the like terms. Remember that the coefficient of [latex]x[/latex] is [latex]1\left(x=1x\right)[/latex].
[latex]\begin{array}{l}3x^{2}+\left(-3+1+5\right)x+1\\3x^{2}+\left(3\right)x+1\end{array}[/latex]
Write the sum as the new coefficient:
[latex]3x^2+3x+1[/latex]
Answer
[latex]3x^{2}+3x+x+1+5x=3x^{2}+3x+1[/latex]
Try It
Simplify [latex]-5x^{2}+3x-x-1+5x[/latex] by combining like terms.
Try It
Simplify [latex]7x^{2}+3x^2-4x-7+5x-3[/latex] by combining like terms.
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 monomials.
ADDING AND SUBTRACTING MONOMIALS
- Combine like terms.
- Simplify and write in standard form.Pay attention to signs when adding or subtracting monomials. In the example below, we are subtracting a monomial with a negative coefficient.
example
Simplify: [latex]11n-\left(-8n\right)[/latex]
Solution
| [latex]11n-\left(-8n\right)[/latex] | |
| Write subtraction of a negative as addition of a positive | [latex]11n+(+8n)[/latex] |
| Combine like terms. | [latex]19n[/latex] |
try it
In order to add monomials, they must be like terms. If 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.
example
Simplify: [latex]{a}^{2}+4{b}^{2}-7{a}^{2}[/latex]
Solution
| [latex]{a}^{2}+4{b}^{2}-7{a}^{2}[/latex] | |
| Reorder the terms. | [latex]{a}^{2}-7{a}^{2}+4{b}^{2}[/latex] |
| Combine like terms. | [latex]-6{a}^{{}^{2}}+4{b}^{2}[/latex] |
Remember, [latex]-6{a}^{2}[/latex] and [latex]4{b}^{2}[/latex] are not like terms. The variables are not the same.
In this example, we switched the order of two of the monomials so that like terms were written next to each other. This is an example of the commutative property of addition.
try it
Adding Polynomials
Adding and subtracting polynomials can be thought of as just adding and subtracting multiple monomials i.e., combining like terms. We use both the Commutative and Associative properties to add and subtract polynomials. Using these two properties we can group like terms that can then be added or subtracted.
Example
Add. [latex]\left(3b+5\right)+\left(2b+4\right)[/latex]
Solution
Reorder and regroup
[latex]\left(3b+2b\right)+\left(5+4\right)[/latex]
Combine like terms.
[latex]5b + 9[/latex]
Answer
[latex]\left(3b+5\right)+\left(2b+4\right)=5b+9[/latex]
In this example, we reordered the terms using the commutative property of addition. We also regrouped the terms using the associative property of addition.
When we add polynomials that include negative coefficients, it is important to remember to keep the negative sign with the term it belongs to.
EXAMPLE
Add. [latex]\left(-5x^{2}–10x+2\right)+\left(3x^{2}+7x–4\right)[/latex]
Solution
Collect like terms, making sure to keep the sign of each term.
The polynomial now looks like this, with like terms collected:
[latex]\begin{array}{c}\underbrace{\left(-5x^{2}+3x^{2}\right)}+\underbrace{\left(-10x+7x\right)}+\underbrace{\left(2-4\right)}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x^2\text{ terms }\,\,\,\,\,\,\,\,\,\,\,\,x\text{ terms}\,\,\,\,\,\,\,\,\text{ constants }\end{array}[/latex]
The [latex]x^2[/latex] terms will simplify to [latex]-2x^{2}[/latex] since [latex]-5+3=-2[/latex]
The [latex]x[/latex] will simplify to [latex]-3x[/latex] since [latex]-10+7=-3[/latex]
The constant terms will simplify to [latex]-2[/latex] since [latex]2-4=-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 standard form (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.
Answer
[latex]\left(-5x^{2}-10x+2\right)+\left(3x^{2}+7x-4\right)=-2x^{2}-3x-2[/latex]
Example
Find the sum: [latex]\left(4{x}^{2}-5x+1\right)+\left(3{x}^{2}-8x - 9\right)[/latex].
Solution
Collect like terms:
[latex]\left(4{x}^{2}+3x^2\right )+ \left (-5x-8x\right )+\left (1-9\right)[/latex]
Combine like terms:
[latex]\left(7{x}^{2}\right )+ \left (-13x\right )+\left (-8\right)[/latex]
Simply and remove parentheses:
[latex]7{x}^{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:
Example
Add. [latex]\left(3x^{2}+2x-7\right)+\left(7x^{2}-4x+8\right)[/latex]
Solution
Write one polynomial below the other, making sure to line up like terms.
[latex]\begin{array}{r}3x^{2}+2x-7\\+7x^{2}-4x+8\end{array}[/latex]
Combine like terms, paying close attention to the signs.
[latex]\begin{array}{r}3x^{2}+2x-7\\\underline{+7x^{2}-4x+8}\\10x^{2}-2x+1\end{array}[/latex]
Answer
[latex]\left(3x^{2}+3x-7\right)+\left(7x^{2}-4x+8\right)=10x^{2}-2x+1[/latex]
Sometimes in a vertical arrangement, we 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.
Example
Add. [latex]\left(4x^{3}+5x^{2}-6x+2\right)+\left(-4x^{2}+10\right)[/latex]
Solution
Write one polynomial below the other, lining up like terms vertically.
To keep track of like terms, insert zeros where there aren’t any shared like terms. This is optional, but it may be helpful.
[latex]\begin{array}{r}4x^{3}+5x^{2}-6x+2\\+0\,\,-4x^{2}\,\,+0\,\,+10\end{array}[/latex]
Combine like terms, paying close attention to the signs.
[latex]\begin{array}{r}4x^{3}+5x^{2}-6x+\,\,\,2\\\underline{+0\,\,-4x^{2}\,\,+0\,\,+10}\\4x^{3}\,+\,\,x^{2}-6x+12\end{array}[/latex]
Answer
[latex]\left(4x^{3}+5x^{2}-6x+2\right)+\left(-4x^{2}+10\right)=4x^{3}+x^{2}-6x+12[/latex]
try it
The following video shows more examples of combining like terms by adding polynomials.
Subtracting Polynomials
When we subtract one polynomial from another, we must distribute the subtraction sign by multiplying every term in the polynomial being subtracted by [latex]-1[/latex]. This is equivalent to finding the opposite of the polynomial being subtracted. We 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.
Example
Subtract. [latex]\left(15x^{2}+12x+20\right)–\left(9x^{2}+10x+5\right)[/latex]
Solution
Multiply the polynomial being subtracted by [latex]-1[/latex]. i.e., change the sign of each term in the polynomial [latex]9x^{2}+10x+5[/latex]! All the terms are positive, so they will all become negative.
[latex]\left(15x^{2}+12x+20\right)–\left(9x^{2}+10x+5\right)[/latex]
[latex]\left(15x^{2}+12x+20\right)-9x^{2}-10x-5[/latex]
Reorder to match like terms, remember to check the sign of each term.
[latex]15x^{2}-9x^{2}+12x-10x+20-5[/latex]
Combine like terms.
[latex]6x^{2}+2x+15[/latex]
Answer
[latex]\left(15x^{2}+12x+20\right)-\left(9x^{2}+10x+5\right)=6x^{2}+2x+15[/latex]
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 show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.
Example
Find the difference of [latex]\left(7{x}^{4}-{x}^{2}+6x+1\right)\text{ and }\left(5{x}^{3}-2{x}^{2}+3x+2\right)[/latex]
Solution
[latex]\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)[/latex]
[latex]\begin{array}{ccc}\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)\text{ }\hfill & \text{Distribute -1 to each term in the second polynomial}.\hfill \\ 7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\text{ }\hfill & \text{Reorder to collect like terms}.\hfill \\ 7{x}^{4}-5{x}^{3}+\left(-{x}^{2}+2{x}^{2}\right)+\left(6x - 3x\right)+\left(1 - 2\right)\text{ }\hfill & \text{Combine like terms}.\hfill \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\hfill & \text{Simplify}.\hfill \end{array}[/latex]
Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.
Example
Subtract: [latex]\left(14x^{3}+3x^{2}–5x+14\right)–\left(7x^{3}+5x^{2}–8x+10\right)[/latex]
Solution
Distribute -1 to each term by changing the sign of each term in the polynomial [latex]7x^{3}+5x^{2}–8x+10[/latex]
[latex]\left(14x^{3}+3x^{2}-5x+14\right)-7x^{3}-5x^{2}+8x-10[/latex]
Reorder to put like terms together and combine like terms.
[latex]\begin{array}{c}\underbrace{14x^{3}-7x^{3}}+\underbrace{3x^{2}-5x^{2}}-\underbrace{5x+8x}+\underbrace{14-10}\\=7x^{3}\,\,\,\,\,\,\,\,\,=-2x^{2}\,\,\,\,\,\,\,\,\,\,=3x\,\,\,\,\,\,\,\,\,\,=4\end{array}[/latex]
Write the resulting polynomial with each term’s sign in front.
[latex]7x^{3}-2x^{2}+3x+4[/latex]
Answer
[latex]\left(14x^{3}+3x^{2}-5x+14\right)-\left(7x^{3}+5x^{2}-8x+10\right)=7x^{3}-2x^{2}+3x+4[/latex]
We can also use the vertical approach to keep our terms organized. But we must distribute the subtraction by multiplying the polynomial being subtracted by [latex]-1[/latex] before we add like terms.
Example
Subtract: [latex]\left(14x^{3}+3x^{2}–5x+14\right)–\left(7x^{3}+5x^{2}–8x+10\right)[/latex]
Solution
Reorganizing using the vertical approach.
[latex]14x^{3}+3x^{2}-5x+14-\left(7x^{3}+5x^{2}-8x+10\right)[/latex]
Change the signs, and combine like terms.
[latex]\begin{array}{l}14x^{3}+3x^{2}-5x+14\,\,\,\,\\\underline{-7x^{3}-5x^{2}+8x-10}\\=7x^{3}-2x^{2}+3x+4\,\,\,\end{array}[/latex]
Answer
[latex]\left(14x^{3}+3x^{2}-5x+14\right)-\left(7x^{3}+5x^{2}-8x+10\right)=7x^{3}-2x^{2}+3x+4[/latex]
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.
try it
Example
Subtract [latex]\left({m}^{2}-3m+8\right)[/latex] from [latex]\left(9{m}^{2}-7m+4\right)[/latex].
Solution
When we are asked to subtract A from B, we start with B and subtract A: B – A
[latex]\left(9{m}^{2}-7m+4\right)-\left({m}^{2}-3m+8\right)[/latex] Remove parentheses: Distribute the – sign.
[latex]9{m}^{2}-7m+4-{m}^{2}+3m-8[/latex] Reorder to collect like terms.
[latex]9{m}^{2}-{m}^{2}-7m+3m+4-8[/latex] Simplify by combining like terms.
[latex]8{m}^{2}-4m-4[/latex]
TRY IT
The following video shows more examples of subtracting polynomials.
The next video shows more examples of adding and subtracting polynomials.
Summary
We have seen that subtracting a polynomial is equivalent to adding the opposite of the polynomial being subtracted. This 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 like terms.
