Adding and Subtracting Polynomials

Hi, and welcome to this review of polynomials! Today we are going to dive into the topic of adding and subtracting them.

A polynomial can be monomial such as $3x^2$, binomial such as $3x^2–4x$, or trinomial such as $3x^2–4x+5$.

Polynomials in the Real World

Working with polynomials is not just a skill we need to master for algebra class; we see polynomials used in real-world situations all the time. Economists use polynomials to model economic growth patterns, medical researchers use polynomials to describe growth patterns of bacterial colonies, and rollercoaster designers use polynomials to graph out the design of a ride.

Simplifying Polynomials

But before we jump into the process of adding and subtracting polynomials, it is important to remember how to simplify polynomials. The process for this revolves around the skill of combining like terms. When we combine like terms, we are essentially combining or consolidating all of the terms that have the same variable, raised to the same power.

For example, let’s quickly look at this expression that needs to be simplified.

 
In order to simplify this expression, we need to combine terms that have the same variable raised to the same power. This means we are going to combine all of the terms that contain x2, all of the terms that contain x, and all of the terms that are constants, which means they have no variable.

With each section simplified, we can rewrite our expression as $-7x^2–4x–4$.

Now that we have a solid foundation with the process of simplifying expressions, we can move into the process of adding and subtracting polynomials!

Adding Polynomials

Let’s add these two polynomials together.

 
The first thing we need to do here is simplify like terms from both sides.

Here, we have $x^3$ and $-x^3$ as like terms, $2x^2$ and $-3x^2$ are like terms, and $8x$ and $2x$ are like terms.

Now, we can combine these terms, and the result will be our answer.

$x^3–x^3$ equals zero, so it’s canceled out. $2x^2–3x^2=-x^2$. And $8x+2x=10x$.

Now we can simply rewrite our answer as $-x^2+10x$.

Subtracting Polynomials

Let’s try an example where we have to distribute a negative sign into our second set of grouping symbols.

In this example, we will need to remember that when you subtract a negative from a negative, the minus sign flips and becomes a positive. So, for example, $-6–-9=3$. All right, let’s subtract these two polynomials:

 
First, just like before, let’s combine our like terms so we can simplify them. $-6x^3$ and $2x^3$ are like terms, $5x^2$ and $-4x^2$ are like terms, and -3 and 1 are like terms. $-3x$ will become $3x$ because it is a negative term being subtracted from another expression, and nothing minus a negative becomes a positive.

$-6x^3–2x^3=-8x^3$. $5x^2–-4x^2$ becomes $5x^2+4x^2$ since we’re subtracting a negative, and that gives us $9x^2$. -3 minus 1 is -4. Now, we put all of these pieces together for our final answer: $-8x^3+9x^2+3x-4$.

Writing Polynomials in Proper Form

It is helpful to know that when you write your answer, it is considered to be in “proper form” when the exponents are arranged by descending powers. For example: $x^3$, and then $x^2$, and then $x$, and then your constant.

So if our answer produced $5x+3+8x^2-x^3$, we would want to rearrange this so that it’s written as $-x^3+8x^2+5x+3$.

Practice Problems

Okay, before we go, try out a couple of example problems on your own! Here’s the first one:

 
Pause the video and see what answer you get!

Okay, let’s look at it together. First, you should have combined like terms, so let’s do that.

When we put these pieces together, we get $3+x^2+7x^3+7x$. In proper form, this would be written as $7x^3+x^2+7x+3$.

Let’s try one more! Pause the video and see if you can subtract these two polynomials:

 
Think you got it? Let’s take a look.

First, let’s combine the like terms.

Now, we can put this together to get $22+x+8x^3+6x^2$. In proper form, this would be $8x^3+6x^2+x+22$.

All right, that’s all for this review! Thanks for watching, and happy studying!