Hello in this video we'll be discussing polynomials. We'll talk about what a polynomial is and what it's made up of And we'll also talk about how to evaluate them In general, polynomials are made up of terms. More specifically, polynomials are made up of sums or differences of these terms. These terms are made up of coefficients, the number that's out in front of the variable, variables. And most of the time the variables will have an exponent on them Like I said a polynomial is a sum or difference of these terms. The key though is that in order be a polynomial, all of the variables have to have whole number exponents, and no variables can appear in the denominators. Reading between the lines here this also means that our exponent can't be negative because a negative exponent really indicates that that variable is living in the denominator of a fraction. So to be a polynomial that sum or difference of terms has only whole number positive exponents. Here are some examples. So we have five different examples of polynomials. You'll notice that we have sums we have differences. Again looking specifically at those exponents we have positive whole number exponents…Polynomials can really be broken down into different types. We have trinomials, tri meaning three So they have three terms. We can have binomials by meaning two. and so binomials have two terms And then we can have monomials…meaning we just have one singular term. So we have this variety of of different types of polynomials and these are some specific names for specific types of polynomials based on the number of terms So just a quick aside what does an exponent mean So if we have something like p to the fourth what does that mean Well p to the fourth is really p times p times p times p So the four tells you how many times are you multiplying the base of that term that variable how many times do you multiplying it by itself? So similarly if we had p squared we could write that as p times p And what exponents do for us is really just gives us a shorthand way of writing this. If we have p to the seventh we really don't wanna write p times p times p seven times. So this gives us a shorthand way to write…So now let's talk for just a moment about order of operations. This is something you might be familiar with already You might be familiar with PEMDAS or please excuse my dear aunt Sally is another mnemonic device for us to use as we remember the order of operations. Now the purpose of order of operations is to tell us about how we should go about evaluating, polynomials or functions or equations, should we be evaluating them and in what order? And So the proper order of operations starts with p parentheses. When there are parentheses we work from the inside of the parentheses out. Now something that's a little bit of a read between the lines is this idea that numerators and denominators of fractions when we deal with fractions are treated like expressions in parentheses. So if you have a numerator and a denominator, you treat them as if they're in parentheses, and so they get evaluated first. So start with the numerator, evaluate, then go to the denominator and evaluate, then do the division. After parentheses we go on to exponents or powers So e. So here we evaluate any of those powers those exponents, Then we go on to m and d and m and d are grouped together M stands for multiplication, d stands for division. And we do this just from left to right So how we normally read we just start at the beginning and we work from left to right doing our multiplication or division whichever comes first. Finally we do a and s A stands for addition s stands for subtraction. This is also another one that we do from left to right. So again you might be familiar with Pemdas. one that I used whenever I was in school was please excuse my dear aunt Sally that also helps us to remember these order of operations. So finally for this video we're going to walk through a couple examples of how to evaluate…using order of operations. So here we want to evaluate in brackets…six…and then parentheses x plus one squared, plus three x minus twenty two and then that's all raised to the second power. So as we're thinking about order of operations what I noticed is that we really have kind of two sets of parentheses We have the brackets and then we have the parentheses…We want to evaluate this for x equals two So the very first thing I'm going to do is I'm going to replace all of my x's with twos…Now that I've done that I'm going to think about my parentheses. So again working from the inside out means that I wanna find the innermost set of parentheses…So I'm highlighting that there So I wanna first evaluate two plus one…So two plus one gives us three…Now again we have a second set of parentheses because the brackets are also considered parenthesis here We have that second set but we have to work from the inside out So now we need to focus on what's inside those brackets. As I look I see an exponent. And so that exponent is what I'm going to have to deal with next So let me highlight. So we are going to deal with three raised to the second power…So three squared or three raised to the second power would be three times three which is nine…And then let's bring the rest down…So now we've dealt with exponents that are inside the parenthesis I recognize we still have that exponent on the outside but since it's outside the parenthesis it has to wait. And so now what we're going to do is we're moving on to multiplication slash division. Well we don't have any division here but we do have two bits of multiplication that we have to deal with…The six times nine and the three times two…So like I mentioned previously we're we go from left to right So starting with six times nine we get fifty four…plus three times two gives us six. Minus twenty two and that's all going to be squared…Now that we've done the multiplication and or division step we move on to addition slash subtraction. Here we have both addition and subtraction We're gonna move from left to right. So the first thing that I'm going to do is add fifty four plus six to give me sixty…From here now I'll do the subtraction sixty minus twenty two gives me thirty eight. So finally I have done everything inside of those brackets and so now I can deal with that exponent on the outside. So now I'll take thirty eight and square it So thirty eight times thirty eight gives me one thousand four hundred and forty four And so that would be my final answer here. So again what did we do We used order of operations to evaluate. Let's do one more example…So here we have a fraction. We're going to evaluate x squared plus three x plus six all over x plus six for x equals two So just like with the last one I'm gonna start by plugging in two to all of my x's or replacing all the x's…with twos…Now as we think about order of operation remember that the numerator and denominator are treated as if they're in parentheses. So although we don't have parentheses, we could if it's easier to help you remember, put some parenthesis around the numerator and denominator. So now what I'm going to do is I'm first going to look inside the parenthesis in the numerator and I'm going to start to use order of operations. So, we're inside the parentheses So now I'm gonna move on to exponents. So we're going to have two squared So that will give me four. Plus three times two plus six…all over And as I look at the denominator well here I'm just taking two plus six inside my parentheses So I can automatically do that and get eight…So the numerator still has a little bit more work we have to do because notice that we have some multiplication. So inside of the numerator we need to do three times two. That comes next in order of operations So we'll get four. Plus three times two gives us six plus six all over eight. And then finally with order of operations now we can move on to the addition subtraction. We only have addition here So we are just going to add four plus six plus six to get sixteen…So we'll get sixteen all over eight. So now we've dealt with the parenthesis part and so now we're moving through order of operations of what we have left. so we don't have any exponents now, but we do have some division And so that tells us we'll take sixteen divided by eight to get two So that would be our final answer here. So in this video we talked about what is a polynomial what is it made up of And we also talked about order of operations and how we can apply them so we can evaluate different expressions.