单项式加减法

Adding and Subtracting Monomials of the Same Degree

Recall that the degree of a term is equal to the value of the exponent on its variable. As a refresher:

If a term has multiple variables, the degree is the sum of the exponents.

For example, $$4x^2y$$ has a degree of $3$ because we add the $2$ on the $x$ and the $1$ on the $y$.

If two monomials (single terms) are of the same degree, we can simply add and subtract their coefficients, like this:

$$\begin{align} &-3x^2+5x^2\\ &=(-3+5)x^2\\ &=2x^2 \end{align}$$

And this: $$\begin{align} &4y^3-y^3\\ &=(4-1)y^3\\ &=3y^3 \end{align}$$

This is similar to combining like terms with algebraic equations.

What if the Monomials Aren't of the Same Degree?

If that's the case, then all you can do is simplify the expression as much as possible and then leave it alone. You can't force terms that don't have the same exponents together. Don't do it :)

You might see an example like this one, below. In that case, just simplify the expression as much as you can. $$\begin{align} &7x+2-4x^2+3x-8+x^3\\ &=(7+3)x+(2-8)-4x^2+x^3\\ &=10x-6-4x^2+x^3\\ &=x^3-4x^2+10x-6 \end{align}$$

Here, we've got a cubic function. We can also have quadratic functions like this!

What are monomials and how to add/subtract them $\\$

Recall what we learned about monomials and polynomials. Here is a chart to help you:

Before we get to polynomials, let's first learn how to add and subtract monomials.

Recall the different parts of a monomial:

We can only add or subtract monomials when they have the same variable and exponent. It might help to refresh combining like terms with exponents.

When adding or subtracting monomials, simply add or subtract the coefficients.

Or, think about it this way:

This question is tricky. Although $x^2$ is in both monomials, the second monomial has two variables. Remember, we can only add or subtract monomials with the same variables and exponents.

Multiplication and division have special rules, so don't try this same pattern with those!

(Video) Adding Monomials

by Lorain Digital Academy

This video by Lorain Digital Academy goes over what the different parts of a monomial are and how to add like terms.

$\large{\text{Summary}}$

For two terms to be "like terms", they have to have the same variable and the same exponent. If you need to review these terms, check out Intro to Polynomials.

So, for example, we would consider $7x^2$ and $-x^2$ to be like terms since they both have the same variable ($x$) and the same exponent ($2$). However, we would consider $7x^2$ and $-y^2$ to not be like terms since the variable is different.

Now, to add/subtract two monomials which are like terms, we simply add/subtract their coefficients. Going to the example above, we can simplify $7x^2-x^2 = (7-1)x^2 = 6x^2$ since $7x^2$ and $-x^2$ are like terms.