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Adding and Subtracting Polynomials

One of the first things we learn is addition of integers. At this point in your math career, if I asked "What is $1+2$?", you would be able to instinctively answer $3$. This skill is extended when we add two different monomials.

$$x+2x = 3x$$

We add $x$'s together the same as we add integers. Just make sure that any terms you add together have the exact same type of variable—otherwise addition won't work right. For example, the following equations hold true:

$$\begin{align}x+3=&4x\\ 2y+8y &=10y\\ s^2+s^2+s^2&=3s^2\\ 8yz-7yz&=yz\\ t-3t&=-2t \end{align}$$

The above equations all work because they have the exact same variables. Let's see some that don't work.

$$\begin{align}xy-x&=y\\ x^2-2x&=-2x\\ 5+3x&=8x\end{align}$$

None of these equations hold true because the variables don't match through the entire problem.

Adding Polynomials

We call what we were practicing above collecting like terms. We put all the $x$'s together, the integers together, as well as any other type of thing. For instance:

$$\begin{gather} 2 + x + 2 + y + 2 + 5x + 8y + 10\\ (1+5)x + (1+8)y + (2+2+2+10)\\ 6x + 9y +16 \end{gather}$$

In order to sum that polynomial, we collect like terms by adding up all of the coefficients, as is seen in the example above. Coefficients are the numbers in front of the variables. For instance, the coefficient of $3x$ is $3$ and the coefficient of $x$ is $1$.

In many math problems, polynomials are presented as a series of different powers of one variable. For instance, an expression like

$$3x^3-2x^2+x-5$$

is very common.

Another thing to keep in mind is that subtraction works exactly like addition. Just as you would expect

$$3x -2x = x$$

Lastly, when a coefficient equals zero, the term is canceled out. So

$$2x-2x=0x=0$$

This is because no matter what the variable $x$ equals, $0x$ will equal $0$ since $0$ times any number is $0$.

Other Properties of Adding Polynomials

Adding polynomials is said to be commutative since

$$2x + x = x + 2x = 3x$$

This means that no matter what order you add them in, you get the same result! Polynomials are also associative which means that

$$(2x+3x)+x = 2x + (3x+x) = 6x$$

This means that no matter where you put parentheses the outcome of adding polynomials is the same!

Adding or Subtracting Polynomials with Multiple Variables

When adding or subtracting polynomials, remember these three steps:

Step One: Distribute (if subtracting) because of the order of operations

Step Two: Group like terms

Step Three: Combine like terms

This last step is really important. When you are adding or subtracting polynomials with multiple variables, the most important thing to remember is that in order to combine like terms, each term must have the same variable(s) with the same exponent(s).

Remember to be extra careful in identifying like terms when more than one variable is involved!

Here is a graphic explaining the steps to adding and subtracting polynomials, along with an example:

Let's do more practice!

Solution

Let's go through each of the three steps to solve this problem! $$(3x^2y^2z^2 + 15x + 2xy+xz^2) + (4x^2y^2z^2 - 5y) - (3x + x^2z - 2yx)$$

Step One: Distribute (if subtracting)

We are subtracting one polynomial, $3x + x^2z - 2yx,$ in this expression, so we need to distribute the negative accordingly. $$(3x^2y^2z^2 + 15x + 2xy+xz^2) + (4x^2y^2z^2 - 5y) - 3x - x^2z + 2yx$$

Step Two: Group like terms

Be extra careful that the terms you are grouping are actually like terms! $$3x^2y^2z^2 + 4x^2y^2z^2 + 15x - 3x + 2xy+ 2yx +xz^2 - 5y - x^2z$$

($2xy+ 2yx$ are like terms, because they have the same variables with the same corresponding exponents ($1$), just in a different order. Order of the exponents does not matter!)

Step Three: Combine like terms

Now, just add or subtract! $$\boxed{7x^2y^2z^2 + 12x+ 4xy +xz^2 - 5y - x^2z}$$

(Video) Add and Subtract Polynomials with Several Variables

by Alane Tetoni

The video by Alane Tetoni shows how to add and subtract polynomials with multiple variables.

$\large{\text{Summary}}$

Recall that two terms are alike if they have the same variable raised to the same power. When dealing with multiple variables, the same thing is true.

They each have to have the same variables. $$\begin{align} 4{\color{Blue}{x}}{\color{#8925AE}{y}}\quad\text{and}\quad7{\color{#8925AE}{y}}{\color{Blue}{x}}:\quad&\text{Like Terms}\\ 8{\color{Blue}{x}}{\color{#8925AE}{y}}\quad\text{and}\quad2{\color{Blue}{x}}{\color{#29A603}{z}}:\quad&\text{Not Like Terms} \end{align}$$

And their variables have to be raised to the same powers. $$\begin{align} x^{\color{Blue}{2}}y^{\color{#29A603}{5}}\quad\text{and}\quad 3y^{\color{#29A603}{5}}x^{\color{Blue}{2}}:\quad&\text{Like Terms}\\ 2x^{\color{Blue}{2}}y\quad\text{and}\quad9xy^{\color{#29A603}{2}}:\quad&\text{Not Like Terms} \end{align}$$

Let's go through some examples. $$(6xy^2-5xy+7)+(9xy^2+2xy-6)$$

We add these just like we would single variable polynomials. Drop the parentheses, try combining like terms, and combine their coefficients. $$\begin{align} &(6{\color{#8925AE}{xy^2}}-5{\color{Blue}{xy}}+7)+(9{\color{#8925AE}{xy^2}}+2{\color{Blue}{xy}}-6)\\ &=6{\color{#8925AE}{xy^2}}-5{\color{Blue}{xy}}+7+9{\color{#8925AE}{xy^2}}+2{\color{Blue}{xy}}-6\\ &=(6{\color{#8925AE}{xy^2}}+9{\color{#8925AE}{xy^2}})+(-5{\color{Blue}{xy}}+2{\color{Blue}{xy}})+(7-6)\\ &=15{\color{#8925AE}{xy^2}}-3{\color{Blue}{xy}}+1 \end{align}$$

The next example uses subtraction. $$(5x^3-9x^2y+3xy^2-4)-(3x^3-6x^2y-2xy^2+3)$$

We subtract these just like we would single variable polynomials. Distribute the minus sign, drop the parentheses, group like terms, and combine their coefficients. $$\begin{align} &(5{\color{#8925AE}{x^3}}-9{\color{Blue}{x^2y}}+3{\color{#29A603}{xy^2}}-4)-(3{\color{#8925AE}{x^3}}-6{\color{Blue}{x^2y}}-2{\color{#29A603}{xy^2}}+3)\\ &=(5{\color{#8925AE}{x^3}}-9{\color{Blue}{x^2y}}+3{\color{#29A603}{xy^2}}-4)+(-3{\color{#8925AE}{x^3}}+6{\color{Blue}{x^2y}}+2{\color{#29A603}{xy^2}}-3)\\ &=5{\color{#8925AE}{x^3}}-9{\color{Blue}{x^2y}}+3{\color{#29A603}{xy^2}}-4-3{\color{#8925AE}{x^3}}+6{\color{Blue}{x^2y}}+2{\color{#29A603}{xy^2}}-3\\ &=(5{\color{#8925AE}{x^3}}-3{\color{#8925AE}{x^3}})+(-9{\color{Blue}{x^2y}}+6{\color{Blue}{x^2y}})+(3{\color{#29A603}{xy^2}}+2{\color{#29A603}{xy^2}})+(-4-3)\\ &=2{\color{#8925AE}{x^3}}-3{\color{Blue}{x^2y}}+5{\color{#29A603}{xy^2}}-7 \end{align}$$

One final example. $$(x^3-y^3)-(-4x^3-x^2y+xy^2+3y^3)$$

This time I won't color the different terms. But do take note that there are $4$ different terms. $$\begin{align} &(x^3-y^3)-(-4x^3-x^2y+xy^2+3y^3)\\ &=(x^3-y^3)+(4x^3+x^2y-xy^2-3y^3)\\ &=x^3-y^3+4x^3+x^2y-xy^2-3y^3\\ &=(x^3+4x^3)+(-y^3-3y^3)+x^2y-xy^2\\ &=5x^3+x^2y-xy^2-4y^3 \end{align}$$