Polynomials Intro - Online Tutor, Practice Problems & Exam Prep

Polynomials are algebraic expressions with variables raised to positive whole number exponents. They can be classified as monomials, binomials, or trinomials based on the number of terms. To express polynomials in standard form, arrange terms in descending order of their degree, which is determined by the highest exponent. When adding or subtracting polynomials, combine like terms carefully, especially when distributing negative signs. Understanding coefficients, constants, and the structure of polynomials is essential for further operations and simplifications.

Introduction to Polynomials

Everyone, so in this video, we're going to start talking about polynomials. We're going to be working with polynomials a lot later on in the course. We'll be doing things like adding, subtracting, and multiplying them. So I wanted to give you a brief introduction as to what they are in this video, and it turns out that when we looked at algebraic expressions, we've already seen them before, so we're just going to go ahead and give them a name. So let's get started. A polynomial, by definition, is an algebraic expression. It's a type of algebraic expression where variables have only whole number exponents, and I actually want to say that it's positive whole number exponents. In other words, when we saw algebraic expressions, we saw something like this where we have 6x^3 and 3x^2 and 5x. All of these things here, if you look at the exponents, they all have positive whole numbers. We have 3, 2, and 5. When it doesn't have an exponent, it just means sort of like an invisible one that's here. All of these things have positive whole number exponents, so this is the definition of a polynomial. So we've seen these types of expressions before. Now we can just call them polynomials. Alright?

So, one of the things I want to mention here is that your exponents can't have negatives or fractions in them. So, for example, when we talked about exponents, we saw some expressions that kind of look like this, like 2x^{-3}. This is not a polynomial because it has a negative exponent. So that's really the whole thing. Always just look at the exponent, make sure it's a positive whole number. Alright. So this expression is an example of a polynomial, but you may see your books refer to these things as monomials, trinomials, and binomials. And, really, the whole thing comes down to the prefix. So the word the prefix monomial, mono means 1. So in other words, this is just a polynomial with one term. The prefix bi in binomial, think bicycle, means 2, and the prefix tri, like think tricycle, means 3 terms. Right? So the whole idea is that a polynomial is kind of like an umbrella term. Polynomial is, like, all-encompassing. But if it has one term, it's a monomial. So, for example, if I only just had the 6x^3, that is a monomial, but if I had the 6x^3 and the 3x^2, and if I had that expression, that's a binomial. But if I had the whole entire thing over here, that's a trinomial, and all of these things are polynomials. Alright? So that's kind of like the umbrella term. Alright.

So the very first types of problems that you might see are just actually figuring out if an expression is a polynomial. Because if so, we're going to have to do things like add, subtract, or multiply them or something like that. So that's what we're going to do in this example problem. And if so, we're going to identify what type it is. If it's a binomial, a monomial, or a trinomial, or whatever. Let's go ahead and get started with the first problem here. We have \frac{3}{4}x + x^3. Now remember, the definition of a polynomial is that we look at the exponents and it has to be a positive whole number. So let's take a look at the different parts of this expression. We have \frac{3}{4}x. So this \frac{3}{4} does that break the rule? Well, remember, the rule only applies to the exponent. It mentions nothing about the terms that are attached to the variables. So in other words, we can have negatives and fractions as numbers, we just can't have them as exponents. So \frac{3}{4}x means \frac{3}{4}x^1, and that's a positive whole number exponent, and x^3, that's also a positive whole number exponent. So in other words, this thing has whole number exponents and this definitely is a polynomial. So how many terms does it have? Well, remember from algebraic expressions, terms are just separated by the pluses and minus signs in your algebraic expressions, so there are 2 terms here. So there's \frac{3}{4}x and the x^3. So if it's a 2-term polynomial, what type is it? Is it a mono, bi, or trinomial? Or a bi means 2, so this is a binomial. Alright. Pretty straightforward. Let's take a look at the second one here. The second one is 5y^{-1}. So what happens? Is this a polynomial? Well, you might think this is just y to the first power, but remember, this is a fraction. And remember from rules of exponents, one way we could rewrite this expression is we could say that this is 5y^{-1}. So because this has negative exponents, this is not a positive whole number, this is not a polynomial. So it turns out this expression is not a polynomial. Even though it only has one term, it actually doesn't fit the definition of a polynomial, so it's none. Alright? So these types of expressions are not polynomials. Alright.

So now let's move to the last one here. We have 2x^3y^2. So in this situation here, we actually have an expression with 2 variables. We've got an x and we've got a y. Is this a polynomial? Well, remember the rule says nothing about having only one type of variable. It only just says that the exponents have to be positive whole numbers. In this case, we have x to the third power, that's a positive whole number, and y to the second. Those are both positive whole numbers. This is a polynomial. So how many terms does it have? Well, in this case, I don't have any pluses or minus signs. Everything's all just sort of multiplied together.

Standard Form of Polynomials

Now that we've seen the basics of polynomials, a lot of problems will ask you to simplify polynomials and write them in a specific form called standard form. That's what I'm going to show you how to do in this video. I'll show you how to write polynomials in standard form. And, really, it just all comes down to the order of the terms in the polynomial. Let's go ahead and take a look here. So, what standard form actually means is that all the terms of your polynomials should be written in descending or decreasing order of exponents. What does that mean? If you look at this expression or this polynomial, 3x2+5x+4, notice how the exponents actually keep decreasing. First, we have an x2. Then we have a 5x1. Right? There's an invisible one there. And then you have +4. And one way you can think of 4, remember, is that you have x0, and x0 is just 1. So it's kind of weird, but, basically, you can see here that the order of the exponents keeps on decreasing: 2 to 1 to then 0. And the other thing is that all like terms have to be combined. In other words, you can't simplify the expression any further. So, if you look at this expression, I can't combine anything with the 3x2 or the 5x or the 4. So it turns out that this expression is already in standard form. Alright? A couple of other things that you should know. One thing is called the degree of the expression, and that's basically just the highest exponent of the variable that you see in the polynomial. So, in other words, the highest exponent that we see that's attached to a variable is the x2. It's the 2. So that means that this is called a second order or a second-degree polynomial. And last but not least, there's just a new vocabulary word. We know that the numbers not attached to variables or by themselves are called constants. We know numbers in front of variables multiplying them are called coefficients, and there's a special name for the number that goes way out in the front that gets attached to the variable of the highest exponent, and that's called the leading coefficient. It's like the number that leads the entire expression. Alright? That's a leading coefficient. That's basically it. So lots of problems are going to ask you to now write expressions in standard form, and that's what we're going to do in this problem. We're going to identify the degree and the leading coefficients. Let's get started here with this first expression. So in other words, we have 12x1 over here plus x3. So we have to write this in standard form, and that means that we have to write it in decreasing order of exponents or descending order. So we have an x3 and an x1. That's backwards, so I just have to flip the two expressions. And when you move expressions around, just be very careful what happens to the signs. In other words, I'm going to rewrite this as x3+12x. Alright? Because it's basically just the x1. So does this have decreasing order of exponents? Yes. It does. And are all the like terms combined? I can't combine anything with the x3 and the one-half x, so this definitely is a simplified expression. So it's in standard form. So what's the degree? The degree is really just the highest exponent that gets attached to a variable, and that one is 3. So in other words, the degree over here is 3. And what about the leading coefficients? Well, the leading coefficient is the number that gets attached to that variable with the highest exponent. And over here, what you'll see is that, basically, there wasn't a variable, but remember, there's always kind of like an invisible one there if you don't see a number. So the leading coefficient in this case is actually just 1. Okay? So let's take a look at this expression; it was a little bit more complicated. There are more terms. Notice that there are some terms with x. There are some with x2, and there are actually just some constants in here. So I have to write them in descending order, but be very careful when you do this because you basically have to keep track of the signs. So, in other words, I have a -3x2 and an x2. So I'm going to move that. I have a 5x and a 2x. I'm going to move that, and I have a -7. So I want to make sure that all the x2s go first. So this is going to be -3x2+x2. Then I'll have the +5x+2x, and then I have the -7. So when you pick these numbers up around and move them, remember that you're always doing this with the sign that goes in front of them. Alright? So now what happens is we have descending order of exponents. I have exponents 2 and then 1, and then I have 0 over here. So this definitely has descending order, but it's not as simplified as it could be because I could still combine all the like terms. So that's what I have to do in the second step. So if I combine this expression over here, -3x and 1x, it's kind of like -3apples and 1apple. This actually just becomes -2apples. What happens to the 5 and the 2? That becomes the 7x, and then the -7 just becomes -7. So now all the like terms have been combined, and this definitely now is in standard form. So what's the degree of this polynomial? What's the highest variable highest exponent of a variable that we see? It's just the 2 over here. Alright? So in other words, we have 2. And then what's the leading coefficient? What's the number that goes in front of that term? It's actually just this -2 over here. So that's the -2. Alright? So, really, that's all. That's it for this one, folks. Let me know if you have any questions, and we'll see you in the next video.

Adding and Subtracting Polynomials

Welcome back, everyone. So now that we've seen the basics of polynomials, lots of problems will now ask you to start manipulating them or doing something with them, like operations. So in this video, I'm going to show you how to do adding and subtracting of polynomials. And I'm going to show you that we've already basically done this when we looked at algebraic expressions. So there's really not a whole lot that's new here. Let's just go ahead and jump right into it. So when we had algebraic expressions and we wanted to simplify them, the easiest way to do that was to combine the like terms. So, for example, if I had x2+8x, I group the x's with the x's and the constants with the constants, and I basically end up with something like this. And the idea here is that these things are actually just polynomials if you just look at them. So we've basically already added polynomials before. So the whole thing is just like algebraic expressions. The way we add and subtract polynomials is where you just perform the operations, and we just combine all the like terms. Alright? So let's just jump right into the first example over here. We have 5x2+2x+3 and x2+7x+8. The idea here is that this thing just looks exactly like this. It's just longer and more complicated. But the idea is that we're just gonna take all the x2 terms, add them and combine them. We'll take the x terms and add them and combine them, and then the constants, if we have any, and add them and combine them. So let's just do it. So if I have 5x2+x2 in this expression, then I combine the like terms, and this becomes 6x2. If I have 2x + 7x, this combines to 9x. And if I have the 3 + 8, this combines down to 11. Alright? So the whole idea is that you're just going to match up these things and then just go ahead and add them. And usually, what you're going to see is that your answers are going to be written in standard form. So in other words, we're going to have decreasing order of exponents and all the like terms combined. But this is how you add and subtract a polynomial. And subtracting a polynomial is very, very similar. So, for example, if I have 3x2+2x+4-5x+10-x2, then I basically just have to do the operation and combine all the like terms. The one thing that is different, though, is that we have to distribute negative signs into parentheses if you have any. It's really important. A lot of students will mess this up, so just be very careful when you do this so you don't get the wrong answer. Basically, you're just going to distribute this negative into everything inside of this parentheses. So let's just do that. So the 3x2+2x+4 doesn't change. Then, we distribute the negative into the 5x, so we get negative 5x. Distribute the negative into the 10, and we get negative 10. And then negative into negative x2, the negatives will cancel. So it's really important there. That's why this step is super important. And now we basically can just drop the parentheses of everything because we're just adding a bunch of terms together. But the idea is the same. We're just going to take the x2's and sort of combine them together, so x2's with x2's, x's with x's, and the constants with the constants. So if you do that, what you end up with getting is 3x2, and then I have plus x2 over here. So that means plus x2 becomes 4x2. This 2x, and then I have a negative 5x. So that will actually be 2-5, which is negative 3x. And then I finally have plus 4 with negative 10. So this will actually just become negative 6. Alright? And so this expression is also in standard form with all the like terms combined. Alright? So that's how to add and subtract polynomials. Just be really careful when you do subtraction because it might get a little tricky. But otherwise, that's it for this one. Thanks for watching.

Perform the indicated operation.

$\left(x^3+3x^2-7x\right)+2\left(x^3-5x^2+9x+4\right)$

• A. $2x^3-2x^2+2x+6$
• B. $3x^3-7x^2+11x+8$
• C. $2x^3-2x^2+2x+4$
• D. $3x^3-2x^2+2x+4$

Perform the indicated operation.

$\left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right)$

• A. $-3x^4+10x^3+7x^2-2x-8$
• B. $-3x^4+17x^3-2x-8$
• C. $-3x^4+17x^2-2x-8$
• D. $-x^4+10x^3-7x^2+14x+2$

Here’s what students ask on this topic:

What is a polynomial in algebra?

A polynomial in algebra is an algebraic expression consisting of variables and coefficients, where the variables are raised to positive whole number exponents. For example, $x^3+\; 2x^2-\; 5x+\; 7$ is a polynomial. The exponents must be non-negative integers, and the coefficients can be any real numbers. Polynomials are classified based on the number of terms they have: monomials (one term), binomials (two terms), and trinomials (three terms).

How do you write a polynomial in standard form?

To write a polynomial in standard form, arrange the terms in descending order of their exponents. For example, the polynomial $3x^2+\; 5x+\; 4$ is already in standard form because the exponents decrease from 2 to 1 to 0. Ensure all like terms are combined, and the polynomial is simplified. The term with the highest exponent is called the leading term, and its coefficient is the leading coefficient.

How do you add and subtract polynomials?

To add or subtract polynomials, combine like terms, which are terms with the same variable raised to the same power. For example, to add $5x^2+\; 2x+\; 3$ and $x^2+\; 7x+\; 8$, combine the $x^2$ terms, the $x$ terms, and the constants: $6x^2+\; 9x+\; 11$. When subtracting, distribute the negative sign before combining like terms.

What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of the variable in the polynomial. For example, in the polynomial $4x^3+\; 3x^2-\; 2x+\; 7$, the highest exponent is 3, so the degree of the polynomial is 3. The degree indicates the polynomial's highest power and helps classify the polynomial's behavior and properties.

What are the leading coefficient and constant term in a polynomial?

The leading coefficient of a polynomial is the coefficient of the term with the highest exponent. For example, in the polynomial $4x^3+\; 3x^2-\; 2x+\; 7$, the leading coefficient is 4. The constant term is the term without a variable, which in this case is 7. The leading coefficient and constant term are important for understanding the polynomial's behavior and graph.