Adding and Subtracting Polynomials

Learning Outcomes

Add and subtract monomials
Add and subtract polynomials

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, $3x^{2}$ and $-5x^{2}$ are like terms: They both have $x$ as the variable, and the exponent is $2$ for each. However, $3x^{2}$ and $3x$ are not like terms, because their exponents are different.

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

Term	Like Terms	UNLike Terms
$a$	$3a, \,\,\,-2a,\,\,\, \frac{1}{2}a$	$a^2,\,\,\,\frac{1}{a},\,\,\, \sqrt{a}$
$a^2$	$-5a^2,\,\,\,\frac{1}{4}a^2,\,\,\, 0.56a^2$	$\frac{1}{a^2},\,\,\,\sqrt{a^2},\,\,\, a^3$
$ab$	$7ab,\,\,\,0.23ab,\,\,\,\frac{2}{3}ab,\,\,\,-ab$	$a^2b,\,\,\,\frac{1}{ab},\,\,\,\sqrt{ab}$
$ab^2$	$4ab^2,\,\,\, \frac{ab^2}{7},\,\,\,0.4ab^2,\,\,\, -a^2b$	$a^2b,\,\,\, ab,\,\,\,\sqrt{ab^2},\,\,\,\frac{1}{ab^2}$

Example

Which of these terms are like terms?

$7x^{3}, 7x, 7y, -8x^{3}, 9y, -3x^{2}, 8y^{2}$

Show Solution

Like terms must have the same variables, so first identify which terms use the same variables.

$\begin{array}{l}x:7x^{3}, 7x, -8x^{3}, -3x^{2}\\y:7y, 9y, 8y^{2}\end{array}$

Like terms must also have the same exponents. Identify which terms with the same variables also use the same exponents.

The x terms $7x^{3}$ and $-8x^{3}$ have the same exponent.

The y terms $7y$ and $9y$ have the same exponent.

Answer

$7x^{3}$ and $-8x^{3}$ are like terms.

$7y$ and $9y$ are like terms.

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.

$2\left(3+6\right)=2\left(3\right)+2\left(6\right)$

Both expressions equal $18$ . 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.

Example

Simplify $3x^{2}-5x^{2}$ .

Show Solution

3 x^{2}

$3x^{2}$ and

5 x^{2}

$5x^{2}$ are like terms.

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

We can rewrite the expression as the product of the difference.

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

Calculate $3–5$ .

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

Write the difference of $3 – 5$ as the new coefficient.

Answer

$3x^{2}-5x^{2}=-2x^{2}$

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.

Example

Simplify $6a^{4}+4a^{4}$ .

Show Solution

Notice that both terms have a number multiplied by

a^{4}

$a^{4}$ . This makes them like terms.

$6a^{4}+4a^{4}$

Combine the coefficients, $6$ and $4$ .

$\left(6+4\right)\left(a^{4}\right)$

Calculate the sum.

$\left(10\right)\left(a^{4}\right)$

Write the sum as the new coefficient.

Answer

$6a^{4}+4a^{4}=10a^{4}$

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.

Example

Simplify $3x^{2}+3x+x+1+5x$

Show Solution

First identify which terms are like terms: only

3 x

$3x$ ,

x

$x$ , and

5 x

$5x$ are like terms.

$3x$ , $x$ , and $5x$ are like terms.

Use the commutative and associative properties to group the like terms together.

$\begin{array}{l}3x^{2}+3x+x+1+5x\\3x^{2}+\left(3x+x+5x\right)+1\end{array}$

Add the coefficients of the like terms. Remember that the coefficient of x is $1\left(x=1x\right)$ .

$\begin{array}{l}3x^{2}+\left(3+1+5\right)x+1\\3x^{2}+\left(9\right)x+1\end{array}$

Write the sum as the new coefficient.

Answer

$3x^{2}+3x+x+1+5x=3x^{2}+9x+1$

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.

How To: Given multiple polynomials, add or subtract them to simplify the expressions

Combine like terms.
Simplify and write in standard form.

example

Add: $17{x}^{2}+6{x}^{2}$

Solution

	$17{x}^{2}+6{x}^{2}$
Combine like terms.	$23{x}^{2}$

try it

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

example

Subtract: $11n-\left(-8n\right)$

Show Solution

Solution

	$11n-\left(-8n\right)$
Combine like terms.	$19n$

try it

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.

example

Simplify: ${a}^{2}+4{b}^{2}-7{a}^{2}$

Show Solution

Solution

	${a}^{2}+4{b}^{2}-7{a}^{2}$
Combine like terms.	$-6{a}^{{}^{2}}+4{b}^{2}$

Remember, $-6{a}^{2}$ and $4{b}^{2}$ are not like terms. The variables are not the same.

try it

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.)

Example

Add. $\left(3b+5\right)+\left(2b+4\right)$

Show Solution

Regroup

$\left(3b+2b\right)+\left(5+4\right)$

Combine like terms.

$5b + 9$

Answer

$\left(3b+5\right)+\left(2b+4\right)=5b+9$

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

Add. $\left(-5x^{2}–10x+2\right)+\left(3x^{2}+7x–4\right)$

Show Solution

Collect like terms, making sure you keep the sign on each term. For example, when you collect the $x^2$ terms, make sure to keep the negative sign on $-5x^2$ .

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 $x^2$ terms would be grouped as $\left(3x^{2}-5x^{2}\right)$ . 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:

$\begin{array}{c}\underbrace{\left(3x^{2}-5x^{2}\right)}+\underbrace{\left(7x-10x\right)}+\underbrace{\left(2-4\right)}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x^2\text{ terms }\,\,\,\,\,\,\,\,\,\,\,\,x\text{ terms}\,\,\,\,\,\,\,\,\text{ constants }\end{array}$

The $x^2$ terms will simplify to $-2x^{2}$

The $x$ will simplify to $-3x$

The constant terms will simplify to $-2$

Rewrite the polynomial with it’s simplified terms, keeping the sign on each term.

$-2x^{2}-3x-2$

As a matter of convention, we write polynomials in descending order based on degree. Notice how we put the $x^2$ term first, the $x$ term second and the constant term last.

Answer

$\left(-5x^{2}-10x+2\right)+\left(3x^{2}+7x-4\right)=-2x^{2}-3x-2$

Example

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

Show Solution

Solution

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

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. $\left(3x^{2}+2x-7\right)+\left(7x^{2}-4x+8\right)$

Show Solution

Write one polynomial below the other, making sure to line up like terms.

$\begin{array}{r}3x^{2}+2x-7\\+7x^{2}-4x+8\end{array}$

Combine like terms, paying close attention to the signs.

$\begin{array}{r}3x^{2}+2x-7\\\underline{+7x^{2}-4x+8}\\10x^{2}-2x+1\end{array}$

Answer

$\left(3x^{2}+3x-7\right)+\left(7x^{2}-4x+8\right)=10x^{2}-2x+1$

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.

Example

Add. $\left(4x^{3}+5x^{2}-6x+2\right)+\left(-4x^{2}+10\right)$

Show Solution

Write one polynomial below the other, lining up like terms vertically.

To keep track of like terms, you can insert zeros where there aren’t any shared like terms. This is optional, but some find it helpful.

$\begin{array}{r}4x^{3}+5x^{2}-6x+2\\+0\,\,-4x^{2}\,\,+0\,\,+10\end{array}$

Combine like terms, paying close attention to the signs.

$\begin{array}{r}4x^{3}+5x^{2}-6x+\,\,\,2\\\underline{+0\,\,-4x^{2}\,\,+0\,\,+10}\\4x^{3}\,+\,\,x^{2}-6x+12\end{array}$

Answer

$\left(4x^{3}+5x^{2}-6x+2\right)+\left(-4x^{2}+10\right)=4x^{3}+x^{2}-6x+12$

try it

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

SCale with a(b+c) on one side and ab+ac on the other adn an equal sign in between the two sides of the scale 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 $−3$ , and changing the sign of $−3$ gives $3$ . Just as changing the sign of a number is found by multiplying the number by $−1$ , we can change the sign of a polynomial by multiplying it by $−1$ . Think of this in the same way as you would the distributive property. You are distributing $−1$ to each term in the polynomial. Changing the sign of a polynomial is also called finding the opposite.

Example

Find the opposite of $9x^{2}+10x+5$ .

Show Solution

Find the opposite by multiplying by

- 1

$−1$ .

$\left(-1\right)\left(9x^{2}+10x+5\right)$

Distribute $−1$ to each term in the polynomial.

$\left(-1\right)9x^{2}+\left(-1\right)10x+\left(-1\right)5$

Your new terms all have the opposite sign:

$\begin{array}{c}\left(-1\right)9x^{2}=-9x^{2}\\\text{ }\\\left(-1\right)10x=-10x\\\text{ }\\\left(-1\right)5=-5\end{array}$

Now you can rewrite the polynomial with the new sign on each term:

$-9x^{2}-10x-5$

Answer

The opposite of $9x^{2}+10x+5$ is $-9x^{2}-10x-5$

You can also write:

$\left(-1\right)\left(9x^{2}+10x+5\right)=-9x^{2}-10x-5$

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

Find the opposite of $3p^{2}–5p+7$ .

Show Solution

Find the opposite by multiplying by

- 1

$-1$ .

$\left(-1\right)\left(3p^{2}-5p+7\right)$

Distribute $-1$ to each term in the polynomial by multiplying each coefficient by $-1$ .

$\left(-1\right)3p^{2}+\left(-1\right)\left(-5p\right)+\left(-1\right)7$

Your new terms all have the opposite sign:

$\begin{array}{c}\left(-1\right)3p^{2}=-3p^{2}\\\text{ }\\\left(-1\right)\left(-5p\right)=5p\\\text{ }\\\left(-1\right)7=-7\end{array}$

Now you can rewrite the polynomial with the new sign on each term:

$-3p^{2}+5p-7$

Answer

The opposite of $3p^{2}-5p+7$ is $-3p^{2}+5p-7$ .

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.

Example

Subtract. $\left(15x^{2}+12x+20\right)–\left(9x^{2}+10x+5\right)$

Show Solution

Change the sign of each term in the polynomial

9 x^{2} + 10 x + 5

$9x^{2}+10x+5$ ! All the terms are positive, so they will all become negative.

$\left(15x^{2}+12x+20\right)-9x^{2}-10x-5$

Regroup to match like terms, remember to check the sign of each term.

$15x^{2}-9x^{2}+12x-10x+20-5$

Combine like terms.

$6x^{2}+2x+15$

Answer

$\left(15x^{2}+12x+20\right)-\left(9x^{2}+10x+5\right)=6x^{2}+2x+15$

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.

Example

Find the difference.

$\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)$

Show Solution

$\begin{array}{ccc}7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\text{ }\hfill & \text{Distribute}.\hfill \\ 7{x}^{4}-5{x}^{3}+\left(-{x}^{2}+2{x}^{2}\right)+\left(6x - 3x\right)+\left(1 - 2\right)\text{ }\hfill & \text{Combine like terms}.\hfill \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\hfill & \text{Simplify}.\hfill \end{array}$

Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.

Example

Subtract. $\left(14x^{3}+3x^{2}–5x+14\right)–\left(7x^{3}+5x^{2}–8x+10\right)$

Show Solution

Change the sign of each term in the polynomial

7 x^{3} + 5 x^{2} - 8 x + 10

$7x^{3}+5x^{2}–8x+10$

$\left(14x^{3}+3x^{2}-5x+14\right)-7x^{3}-5x^{2}+8x-10$

Regroup to put like terms together and combine like terms.

$\begin{array}{c}\underbrace{14x^{3}-7x^{3}}+\underbrace{3x^{2}-5x^{2}}-\underbrace{5x+8x}+\underbrace{14-10}\\=7x^{3}\,\,\,\,\,\,\,\,\,=-2x^{2}\,\,\,\,\,\,\,\,\,\,=3x\,\,\,\,\,\,\,\,\,\,=4\end{array}$

Write the resulting polynomial with each term’s sign in front.

$7x^{3}-2x^{2}+3x+4$

Answer

$\left(14x^{3}+3x^{2}-5x+14\right)-\left(7x^{3}+5x^{2}-8x+10\right)=7x^{3}-2x^{2}+3x+4$

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.

Example

Subtract. $\left(14x^{3}+3x^{2}–5x+14\right)–\left(7x^{3}+5x^{2}–8x+10\right)$

Show Solution

Reorganizing using the vertical approach.

$14x^{3}+3x^{2}-5x+14-\left(7x^{3}+5x^{2}-8x+10\right)$

Change the signs, and combine like terms.

$\begin{array}{l}14x^{3}+3x^{2}-5x+14\,\,\,\,\\\underline{-7x^{3}-5x^{2}+8x-10}\\=7x^{3}-2x^{2}+3x+4\,\,\,\end{array}$

Answer

$\left(14x^{3}+3x^{2}-5x+14\right)-\left(7x^{3}+5x^{2}-8x+10\right)=7x^{3}-2x^{2}+3x+4$

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: $\left(7{u}^{2}-5u+3\right)-\left(4{u}^{2}-2\right)$ .

Show Solution

Solution

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

try it

Exercises

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

Show Solution

Solution

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

TRY IT

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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Unit 3

Learning Outcomes

Combining Like Terms

Example

Answer

Example

Answer

Example

Answer

Example

Answer

Adding and Subtracting Monomials

How To: Given multiple polynomials, add or subtract them to simplify the expressions

example

try it

example

try it

example

try it

Add and Subtract Polynomials

Example

Answer

Example

Answer

Example

Example

Answer

Example

Answer

try it

Find the opposite of a polynomial

Example

Answer

Example

Answer

Subtract polynomials

Example

Answer

Example

Example

Answer

Example

Answer

example

try it

Exercises

TRY IT

Summary

Contribute!