okay five four for math 100 is polynomials in several variables so we have more than just X or Y and our polynomial we have both of them alright so before we actually do what this problem says it says I value eight before we do that let's talk about what it actually means when I see the term 3x cubed Y what that means is 3 times X cubed times y those three terms are those three things are being multiplied together and so that's what XY squared would mean it means x times y squared the 5 y means 5 times y just to make sure that we understand there okay so evaluate means we're actually going to come up with a numerical answer and now they've told us in this problem that they want X to be negative one and they want Y to be 5 so our first step is to substitute those values in place of x and y alright so I've got this 3 it doesn't go anywhere where I saw an X I'm gonna put a negative 1 keep my cubed and whereas all why I'm gonna put a 5 notice off the parenthesis around the number where it belongs okay in my second one I have we're in place of X I put negative 1 in place of Y I put 5 and I still have that squared plus still have this 5 in place of Y I put 5 and then I have plus 6 so let's just make sure we all are completely clear here all right I'm going to draw some lines through this so it's gonna look when I'm done alright so there's the three instead of X I put negative line instead of Y I put five okay I substituted the negative one in place of X the five in place of Y and then I did the same thing on the next one where this said X I put negative one where it said why I put five okay the next one this five I brought it down and then where it said why I put five and then that last number over there that six it just comes down so now I'm able to work my problem I've got everything substituted in where it belongs and so now algebraically I do what it says making sure to use order of operations so order of operations that's the please excuse my dear Aunt Sally the first thing I do is the cubed so this 3 comes down negative 1 cubed is negative 1 okay so that's the first thing I take care of in my second term 5 squared is 25 alright so I've taken care of all of my exponents everything else is just going to come down right and now I'm ready to multiply so 3 times negative 1 times 5 gives me negative 15 negative 1 times 5 gives me negative when do you found out 5 times 5 is plus 25 and then bring down 1 plus 6 now order of operations we add and subtract left to right so first I'm going to do negative 15 minus 25 and I get negative 40 and I'm gonna bring down this 25 plus 6 because I haven't used it the only thing I did right there was I'm put these two together to get the 40 so the twenty five of the six come down all right now I'm actually going to do the negative forty plus twenty five and I get negative fifteen and bring down my plus six so now I have negative fifteen plus six and I end up with negative nine so the solution did this problem is negative nine all right moving on all right so in MyMathLab you're gonna ask you some things about the coefficient and the degree you remember we went over those things in section one let me just look back and make sure we actually went over yeah we went over degree and coefficients and we even made ourselves a little table so we're going to make a table very similar to what we did before right so our next problem is going to say determine the coefficient of each term I'm going to write the directions just so we can see exactly what its gonna look like determine the coefficient of each term the degree of each term and then the last part of the question says and the degree of the polynomial so MyMathLab is really good about asking you multi-step questions so make sure that you answer all of the parts all right and so the polynomial that we're working with here is 8x to the fourth Y to the fifth minus 7x cubed Y squared minus 5x plus 11 now this was a little different than the ones we did in section one because we have more than one variable so we're gonna make a little table just like we did before in our table the first thing I'm going to do is write the term okay my first question was determine the coefficient of the turn so I'm gonna write the entire term alright in the first column my second column is going to be the first answer to the question that's where I'm going to write my coefficients and then the last column is going to be the degree of that term okay and so the last question degree of the polynomial is not in this table at all all right we're still going to answer it but it's not in that table all right so our first term in this particular polynomial is this 8 X to the fourth Y to the fifth our next term make sure when you write down this middle term you take its sign with it same thing with the minus 5 X and then we have to let the 11 at the bottom all right now we're just going to identify the coefficient remember that is the number in front of the variable so our coefficient here is 8 negative 7 negative 5 and 11 the number part of the term all right and then the degree of the term okay the way we found that is that is the sum of the exponents of that particular term so in this first term we see we have in it a 4 and a 5 so the sum of 4 and 5 would give me none okay so I add the 4 in the 5 to get none that would be the degree of the first term the second term I'm going to add the 3 in the - those are my exponents I would get 5 my next one my exponent is a 1 okay I don't have to add it to anything that's just that 1 and then they exponent on the bottom there's not one there's not a variable so my X my degree is 0 okay so we have answered question number one determine the coefficient of each term that's right here okay question number two was on the degree of each term okay awesome did that right here in green the thing that I have not done is I have not answered the degree of the polynomial so I'm going to do that over here on the side the degree of the polynomial doesn't go in a table here okay the degree of my polynomial is the highest of those that are circled in green so the degree of my polynomial is going to be nine right here the degree of my polynomial awesome okay so my math ought is definitely going to ask you a question like that I wanted you to be prepared answer all the parts okay next idea as we are going to add and subtract and we're going to still multiply polynomials okay because the multiplying we've are learned how to do we learned how to add and subtract and now we're just gonna kind of put all of that together add subtract and multiply all together in some problems all right so let's say I ask you to I ask you to add these two polynomials 6x squared Y I'm sorry X Y squared minus 5x y plus seven I'm going to add to that an entire nother pile on another polynomial so 9 X Y squared plus 2xy minus 6 all right so we're adding these we need to identify are there any like terms so X Y squared well here's another X Y squared so those two are like terms 6x squared plus 9x y squared okay and this is where I got that little plus from all right now let's look at five X Y would be a like term with 2xy well don't forget this little sign right here so I've got a minus five X Y and then again I've got this plus 2xy alright and for my last term's I've got this seven and this six they are like terms so I've got plus seven and don't forget I've got this little minus six I do have plush right here but here's what I don't like to do I don't like to do that plus - oh that's just a personal pet peeve of mine so when I see plus and minus beside each other the negative is what we get because positive times negative is negative all right so now I'm ready to combine my like terms alright so 6x y squared plus 9x y squared 15 XY squared notice that my variables do not change just the coefficients change all right and then I have negative 5 XY plus 2xy gives me minus 3 XY all right and then I have plus seven minus six gives me plus one so this would be my final answer to this particular problem 15 XY squared minus three X y plus one we have just added some polynomials that have different variables okay let's do one where we have a subtraction with different variables this is where we have to be careful with our subtraction okay 7x cubed minus 10x squared y plus 2xy squared minus five okay there's our first polynomial and we are going to subtract from it 4x cubed minus 12x squared Y minus 3 XY squared and then plus 5 all right so the first thing I'm going to do is identify any like terms all right so this X cubed and this X cubed so I have 7x cubed minus 4x cubed okay my x squared Y those are like terms so don't forget our little sign right here so I have negative 10 x squared Y about this sign right here that would be a minus oh boy and then I've got this sign right here negative 12x squared Y oh I really really really don't like that - and negative beside each other so again pet peeve of mine so I would not do that negative times negative that actually turns that into plus 12 all right moving on to our next like terms we have - XY squared and a negative 3xy squared so we have plus 2xy squared so this little minus right here and we have that minus there and there it is again minus a negative you can't have two signs beside each other minus a negative is a positive and then we have our last terms that are like terms would be this negative five and this positive that so I have minus five - net let me do that again okay so this five in here in the first one has a minus in front so minus five I still have this negative in the middle so - and this one is a positive five there we go all right now I'm going to combine all of my like terms 7 minus 4 gives me 3x cubed negative 10 plus 12 gives me plus 2 x squared Y now we're on the green ones 2xy squared plus 3x y squared is plus 5 XY squared and then the yellow ones negative 5 minus 5 is negative 10 signs are the same so we add Thank You 13 awesome okay all right so I promise we're gonna do some that have multiplication in them also and some simplifying all right so let's do that let's do X minus y times x squared plus XY plus y squared all right we are multiplying here when that parentheses is beside the other one we are multiplying so big distributive property so we're gonna multiply this first two x times all three of those terms when we're done with that we're gonna multiply this negative y times all three of those terms then we'll worry about like terms all right so x times x squared is x cubed x times X y is plus x squared Y add the exponents on the X's x times y squared would be plus X y squared we're done with all that Green Arrow's okay so now we'll go to negative y times x squared is negative x squared Y again I'm right in that order because I'm I was already riding in the X Y order so I'm gonna continue to do that now negative Y times positive XY gives me negative x y squared negative Y times positive y squared is negative Y cubed all right looking for like terms now be careful because these get a little crazy with x and y all right here the squared is on the X so here's its like term right there squared is on the X all right on this one the squared is on the y squared is on the Y awesome so I think I have them identified x squared X cubed does not have a like term it will come down all right so my x squared Y one of them is positive one of them is negative and I use my zero the ones underlined in blue one of those is positive one of those is negative again that gives me zero and so the only thing I have left is minus y cute so that would be an isolation all right so just be careful with your multiplying make sure you're adding exponents when you multiply and make sure you're looking for like terms at the end of the problem it's simplified if you have any questions just email me and make sure that you stay on pace and try not to get behind our pacing guide so that finishes up Section five four