In the first lesson of this unit, we learned about exponents. For the rest of the unit, however, we're going to focus on polynomials. Let's see if we can figure out what these are.
Before we get started leaning about polynomials, lets actually start off with some shapes. Here's some different shapes and some names of them. What I want you to is to determine which shapes are 2 dimensional and which shapes are 3 dimensional. A 2 dimensional shape is something that's flat that we can pick up. A three dimensional would be something that we can hold like a pen or pencil. What I want you to do is to classify these shapes and let me know if they are 2 dimensional or if they're 3 dimensional. Go ahead.
Well we know that a circle is flattened on a screen, so it must be They're all flat, so they must be 2-dimensional. A cube and a cylinder are definitely 3-dimensional. We can hold a cube in our hand or we can pick up a cylinder, like a cup or a glass. Great work on that quiz.
So, why did we start off with shapes? And why are these different categories of broad category called shapes. We think of shapes as being made up of curves and straight lines connected at points or vertices. Then we looked at how we could find differences between those shapes, two dimensional, and three dimensional. We were wondering, are the shapes flat or not? And, from that single question we were able to make these two categories about our shapes. Well, this is also true about functions. We use functions to describe particular relationships in math. For this unit, we're going to focus on polynomials. Expressions with constants, with linear terms, quadratic terms, or cubic terms are all considered to be part of polynomials. Constants are numbers like 3, negative 4,or 5 halves. Linear terms contain a single variable with an exponent of 1. We don't have to write the 1 as the exponent, because we assume that's the case. A linear term might have constant terms on it or not, but it's the variable with the exponent of 1 that makes it linear. Quadratic polynomials contain an x squared. And, cubic polynomials, they contain a cube, or a third power. For this unit, we won't focus on the graphs of polynomials. We're going to focus on working with them. So adding subtracting multiplying and dividing them. You might be wondering what some of these other functions are. They include square roots and exponentials, but don't worry about these. We'll get to them later.
Lets start with the smimplest polynomial, a monomial. A monomial consists of only one term, and it could contain the product of numbers and variables. When we see a number next to a variable, we assume that there's a multiplication between the two. But, like many things in math, we don't often show it. Here are lot of examples of monomials. I've rearranged our monomials into certain groups. Here's a group of four monomials, and in this group, I have a constant coefficient and a variable. The variable is x in three of these, but it doesn't always have to be. In this one, it's y. These monomials are simply just constants. They don't even contain variables. This monomial is negative and it contains variables with exponents. Monomials can be positive or negative, like in the first group. And finally, this last group shows an example where we have a constant coefficient and multiple variables multiplied together. Each of these is an example of a monomial.
Here are some examples of things that are not monomials. What do you think's different here? What's changed? Take a minute to think about it, and then choose which answers explain why these are not monomials. Check all the boxes that apply.
If you said these two, excellent work. For monomials, we said that the exponent needs to be non-negative, so that means the exponent must be positive or zero. Here are two examples where the exponent in the monomial is negative, so these aren't monomials. This term is negative, but the variable, or n, appears in the denominator. Any expression or term with a variable in the denominator isn't a monomial. These two examples are the same, the b appears in the denominator, and the x appears in the denominator, so these can't be monomials. You might not have gotten this reason right, which is totally okay, it turns out that a square root can be written as x to the one half, we haven't even covered this topic yet, so if you got that right, that's amazing. We'll see this fractional exponent, later but I just wanted to preview it now. We do know that monomials can be the product of two variables and that they can be positive or negative, these are the real reasons, why all of these weren't examples of monomials.
It turns out that any monomial that we encounter will be an example of a polynomial. Monomials have one term, and are the products of constant coefficients and variables. Binomials are another type of polynomial. These contain exactly two terms. A trinomial is another example of a polynomial. This one has three terms. I've been throwing this word term around a lot. And a term is simply the product of numbers and variables. This is one term. This is one term. And this is one term. The same is true for these. When we have addition or subtraction between two terms, that's when we call it a binomial. And the same is true in this case. We call this a trinomial. You can remember these terms based on their prefix. Mono means one. Bi means two. And tri means three. That should be pretty familiar. A bicycle has two wheels. A triangle has three sides. Maybe you can think of some other words that start with these prefixes.
Now that we've covered some terminology with polynomials, let's try using that knowledge with a quiz. I'm going to list some polynomials here. And I want you to tell me if it is a polynomial and if it's a binomial, a trinomial or if it's really not a polynomial, choose other.
Excellent work if you got all these right. We know for the first one, we have three separate terms separated by addition, so that's a trinomial. This second expression is a monomial. We have the product of a constant coefficient, 5, and variables, a cubed, b squared, and c. So just one term. The next expression contains a square root, so it's not a polynomial at all. It's something else. Here we have three separate terms separated by addition and subtraction, just like in the first one. So, it's a trinomial. For 2 m minus 4, I have two separate terms, so that's a binomial. Negative 20 is a monomial. And finally, for this last one, I have a negative exponent on this term, so it must be something else. It's not a polynomial.
We learn the words monomial, binomial, and trinomial. So that way, we could talk about polynomials. I'll be saying those words a lot. So I wanted to make sure you're clear on them. For example, here we have a trinomial and another trinomial. And we're going to try and add these together. Before we get started, I'm going to show you some algebra tiles to help us think through this. Here's a square, and each side of this square is x units in length. I don't know it. But I do know that the area of the square is x times x, or x squared. This rectangle has a legnth of one on one side, and x on the other. So the area of this rectangle would just be x. And finally, this last tiny square, has an area of 1, it's a 1 by 1 square. I drew these shapes out to help us think about like terms. So let me show you what this problem would like, using these shapes. For this first trinomial, I would have three x squared's, four x's, and 3, or three 1's. You can see a similar representation for the second trinomial. So if you want to add these two polynomials together, we want to combine the like terms. To add like terms together, we simply add up the coefficients of the same term. So 3x squared and 2x squared would make 5x squared. 4 x and 2 x would make 6 x's, and second polynomial to get my like terms. Now that we have this, we're ready to combine like terms. Here was the original problem, and if we're just adding we can drop the parentheses and rearrange to find our like terms. So we have 5x squared plus 6x plus 4. This is the trinomial we get when we add these two together.
Here's the sum of two more trinomials. These trinomials are a little bit more complicated because they have multiple variables with exponents. I want you to try this one out. What do you think the sum of this trinomial and this trinomial would be? Enter your answer here. You might stumble a little bit on this one, which is totally okay. I want you to try your best.
When finding the sum of polynomials, we want to identify like terms first. I have 3x squared y squared, and negative 14, x squared, y squared, these are like-terms because they're the same variables with the same exponents. Negative here. These last two terms are tricky, they're actually not like, so I can only list them separately, I won't be able to combine them, or add them. Now I simply count up the number of x squareds y squared that I have, I have positive 3 and negative 14 of them, so that's a total of negative 11. For the next term, I have negative 8xy, and negative 2xy, so that makes negative 10xy altogether. And finally, we just list these terms as sums on the end, because they're not like with anything else in our expression, so here's our answer.
Alright, I know that last one was tough, but try this one out. Try adding these goes in the box.
If you got this answer, excellent work. Here I've gone ahead and drawn the like terms in shapes. The p to the 4ths have a green square, and the p squareds have circles in a lighter color. I also drew triple lines under the constants. I'm showing you this as a method of finding the like terms. Use whatever you feel comfortable doing. These are just tools to help you out. I've rearranged the like terms in this second line. Notice I paid careful attention to keep the sign with its term. If it's positive the sign is positive. And if it's negative, the sign in front of the term still stays negative. Finally, adding our like terms together, we get this result.
Here's the second practice problem. Try adding these two trinomials together and write your answer here. Good luck.
If you got this as your polynomial, great work. This question was pretty tricky. So don't worry if you got it wrong the first time. I've rewritten the like terms in this line. And here's the detail that we really need to watch out for. Here we have 3m cubed. And here, we have 2m squared. Not like terms. Just because you see two trinomials. That doesn't mean you should add each term in its corresponding position together. Sometimes, they're not like terms. When I add my like terms together, I get this result.
Here's our third and final practice. Use your keen attention to detail and your knowledge of like terms to find the sum.
Here are the like terms, and these two aren't like, because they have 2ab squared and 3a squared b. The exponents don't match. After regrouping our like terms together, we add them to get our result. And, remember, these terms aren't like, so we just bring them down into our cell.
Now that you have a handle on adding polynomials, let's try subtracting them. When we subtract polynomials we need to subtract the second quantity. We can think of this problem just like this problem. We know in this problem we don't just subtract the one, we need to subtract a total of six. So we can subtract the 1, subtract the 2 and subtract the 3. This is why these parentheses are so important they indicate that when you subtract a total of 6 and not just a total of 1. We get an answer of 9 here but this isn't so important, whats important is what we did right here. We distributed a negative 1, we're going to use the same line of thinking for tackling this whenever you subtract a quantity, you want to change every single sign because we're multiplying by a negative one. So I multiply each term by negative 1 and really that just changes all their signs. You don't have to show this step in your problem solving, but if it helps, you do it. Here's our expression after we multiply by negative 1. Once we're here we carry out the problem just like addition. We regroup our like terms, and then we combine them x squared plus 2x plus 2. Here's our final polynomial.
Here's your first practice with subtracting polynomials. Write your answer here.
First, we distribute the negative 1 to each term here. So I have negative 5b cubed plus 2b and negative 4. Remember, each of these terms, signs should change. Next, I regroup my like terms. I have 3b cubed and negative 5b cubed here. Negative 7b and positive 2b here. And a positive 9 and negative 4 on the end. And my final answer is negative 2b cubed minus 5b plus 5. Excellent work if you got it.
Try your best on this second practice problem. Good luck.
First, we distribute a negative 1 to each of the terms, and change their signs. Next, we identify the like terms. And finally, we combine the like terms together. Throughout the problem, I just kept my 4m to the 5th. And my negative in my answer. Negative 5m cubed and negative 5m cubed make negative 10m cubed. And positive 3 and negative 3 make 0. So here's my answer.
Here's the last problem for this lesson. Try subtracting this polynomial. Pay close attention to your signs throughout the problem, and write your answer here.
Awesome work if you got this one right. First we distribute a negative 1 to each of the terms, so we change the signs. Next we combine our like terms and regroup them in our next line. 6a to the 4th, and 6a to the fourth make 12a to the 4th. A squared and a cubed just need to be listed down here. Notice that I switched the order of them, that's okay. I know 2 plus 1 is the same thing as 1 plus 2. We can use the commutative property of addition to rewrite our expressions. The order doesn't matter. And finally on the end, we have negative 2. If you're wondering why I switched the order of these, it's because we usually have the powers decrease as we list the terms. We have a to the fourth, a to the cubed, a squared, there's no a term here and then our constant. If you're having any trouble with adding or subtracting polynomials, that's a great time to ask some questions in the forum. We love to hear from you. Maybe you're wondering about like terms, or maybe you're wondering about intergers. If so, let us know.