Combinação de termos

What Are Like Terms?

Like terms have the same variable raised to the same power. The coefficients of these terms do not have to be the same!

For example, $4x^2$ and $3x^2$ are like terms, because they have the same variable, $x,$ that is raised to the same power, $2.$ It doesn't matter that the coefficients of these terms, $4$ and $3,$ are different!

The like terms of $13y^3$ are all the terms that share the same variable, $y,$ raised to the power of $3.$ The only terms that fit this description are $435y^3$ and $y^3.$

Combining Like Terms

To combine like terms, all you have to do is add the coefficients, then multiply that result by unknown value that you're working with!

For example, say we want to evaluate $4x^2 - 3x^2.$ First, realize that: $$4x^2 - 3x^2 = 4x^2 + (-3)x^2$$

The coefficients of the terms in $\boxed{4}x^2 + \boxed{-3}x^2$ are $4$ and $-3.$ To combine these terms and get the coefficient of the final answer, we add $4$ and $-3.$ $$4+(-3)=1$$ Thus, $1$ is the coefficient in our answer. Our last step is to just put this coefficient in front of the unknown value we are working with, which is $x^2.$ So: $$\begin{align} 4x^2-3x^2&=4x^2 + (-3)x^2\\ &=(4+(-3))x^2\\ &=1x^2\\ &=x^2 \end{align}$$

Longer Expressions

Sometimes, you will be given a long expression with multiple powers of variables and asked to combine like terms. One strategy to tackle these types of problems is to group like terms before you combine them. Here's a graphic with an example of how this works:

Let's do another example.

We want to combine like terms to simplify this expression: $$4x^2+6y+3y^3-8y^2+2-4y^3+7x^2-4y^3-17y+y^2-5$$ Step One: Group Like Terms

$$(4x^2+7x^2)+(6y-17y)+(3y^3-4y^3-4y^3)+(2-5)+(-8y^2+y^2)$$

Step Two: Combine Like Terms

$$(4x^2+7x^2)+(6y-17y)+(3y^3-4y^3-4y^3)+(2-5)+(-8y^2+y^2)\\ (4+7)x^2+(6-17)y+(3-4-4)y^3+(2-5)+(-8+1)y^2\\ 11x^2-11y-5y^3-3-7y^2\\ 11x^2-3-11y-7y^2-5y^3, \text{ after some rearranging}$$

So, our simplified expression is: $$11x^2-3-11y-7y^2-5y^3$$

(Video) Combining Like Terms

by Friendly Math 101

Here's a video by Friendly Math 101 that explains how to combine like terms.

$\large{\text{Summary}}$

Like terms are terms with the same variable raised to the same power.

The first example she looks at is $3x-7y+9y+4x$. We start by looking at the first term: $3x$. We want to find all the other terms in our expression that are alike, or in other words have an $x$. $${\color{#29A603}{3x}}-7y+9y{\color{#29A603}{+4x}}$$ Because they're like terms we can combine them. Combine means to add them, but only their coefficients. Combining like terms is really important in math, because it allows you to simplify expressions so they are more manageable. For example, it's really helpful when adding or subtracting polynomials! $$3x+4x=(3+4)x=7x$$ Next, move to the next term in our expression, $-7y$. It's important to note that we keep the sign in front. Now we look for like terms. $$3x{\color{#29A603}{-7y}}{\color{#29A603}{+9y}}+4x$$ Combine these terms. $$-7y+9y=(-7+9)y=2y$$ Our simplified expression now becomes: $$7x+2y$$

$$\begin{align} &8+4x+1-12+5x\\ &= 4x+5x+8+1-12\\ &= 9x+8+1-12\\ &= 9x+9-12\\ &=\boxed{9x-3} \end{align}$$

$$\begin{align} &-2x^3+7x-4x^3+x^2+11x-5\\ &= -2x^3-4x^3+x^2+7x+11x-5\\ &=-6x^3+x^2+7x+11x-5\\ &=\boxed{-6x^3+x^2+18x-5} \end{align}$$

(Video) Simplifying Polynomials

by Math Antics

Here's a great video by mathantics that covers combining like terms.

$\large{\text{Summary}}$

Like terms are terms that have exactly the same variable and power in them—whether that's $x$, $x^3$, $y$, or even no variable!

So, for example, $2x$ and $3x$ would be like terms since they both have the variable $x$ and they're both to the first power.

But $2x$ and $2y$ would not be like terms because they have different variables, $x$ and $y$. Similarly, $2x$ and $2x^2$ wouldn't be like terms because while they have the same variables, the variable is raised to different powers. This can get easily confused with multiplying exponents. It also gets confused with raising a power to a power.

We can only combine terms if they're alike. So we can combine $2x+3x$ but can't combine $2x+2y$. In order to combine, we leave the variable and power alone, and either add or subtract the number out front, the coefficient. Then this is just about adding or subtracting real numbers.

Example 1

Simplify $5x+3x$.

Since these terms are like terms, both have $x$ raised to the first power, we can add them by adding their coefficients. $$\begin{align} &5x+3x\\ &=(5+3)x\\ &=8x \end{align}$$

Example 2

Simplify $3x^2+x^2-7y-3y+7$.

In this problem, only some terms are alike. We have two terms with $x^2$ and two terms with $y$. To simplify, we combine as much as we can. $$\begin{align} &3x^2+x^2-7y-3y+7\\ &=(3+1)x^2-7y-3y+7\\ &=4x^2-7y-3y+7\\ &=4x^2+(-7-3)y+7\\ &=4x^2+(-10y)+7\\ &=4x^2-10y+7\\ \end{align}$$

The like terms are $3y$, $0.5y$, and $-4y$, because they've all got the same variable in them.

$-9y^2$ isn't a like term because of that pesky little exponent. You can't combine terms where the exponents are different!