More examples of addition and subtraction of polynomials | Algebra II | Khan Academy

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In this presentation I want to introduce you to an idea Polynomial equations Perhaps the word sounds fake, but all Meaning is a statement containing a set of variables or constants That are raised to non-zero forces Perhaps this also seems complicated So let me show you an example If I want to give you x ^ 2 + 1, this is it Polynomial equation In fact, these are polynomials because they contain two terms The term polynomial is more general Its content is that you have several limits. Multiple word It means many These are polynomials If you want to say 4x ^ 3 – 2x ^ 2 + 7 Then this is a trinomial I have three limits here Let me give you a precise meaning of what is Polynomial and what is not For example, if I have x ^ -1 / 2 + 1 This is not polynomial This does not mean that you will never see it when Dealing with algebra or mathematics But we will not call this polynomial because it contains A negative and fractional power Or if I want to give you the expression y x square root For y – y ^ 2 Again, this is not polynomial, because it contains A square root, which is basically to raise a number For strength 1/2 So all the foundations of the variables must Be nonzero Again, neither of these is polynomial Now, when we deal with polynomials, we will We get some terms Perhaps it is familiar to you, or it may not be familiar, so I will show it to you now The first term is polynomial degree Polynomial degree And that is our highest exponent In polynomial For example, this polynomial is Third degree polynomial Why? No need to continue writing it Why is considered a polynomial of the third degree? Because the highest exponent is in it X ^ 3 statement From here we obtained a third degree polynomial This is quadratic polynomial This is a second degree phrase Now there is a host of other terms, or words, that We need to know it about polynomials, which are fixed expressions Changing versus, I think you already know There are variable phrases here This is fixed It is fixed And then there is one last part until we explain Polynomial properties is to understand polynomial coefficients Let me write the fifth degree polynomial here And maybe I will write it funky Here I will not put it in the usual order Let’s assume it x ^ 2 – 5x + 7x ^ 5 – 5 Again, this is a fifth degree polynomial Why? Because the highest exponent on the variable here is 5 Here This tells us that it is a fifth degree polynomial And you might say, well, why do we care about that? And at least for me, I care To the degree of polynomial because when you get to numbers The highest degree is what controls In other words, it gets bigger faster, or Decreases faster, depending on if it is The signal is positive or negative in front of it But he will control everything else It gives you meaning to speed Ferry growth is complete in case If it contains a negative coefficient Now I have used the word operand what does that mean? Factor I have used it before, when Linear equations are covered The coefficients are constants Multiplied by changing expressions For example, the operand for this phrase Here is -5 You have to remember that we have -5, so consider An entire -5 parameter The coefficient of this phrase is 7 There are no labs here; It is a constant phrase of -5 Then the coefficient of x ^ 2 is 1 The coefficient is 1 It is not visible You assume it is 1 x x ^ 2 Now, the last thing I want to give you is Idea of \u200b\u200ba typical polynomial shape The typical polynomial form Now, none of this will help you in a polynomial solution But when we talk about polynomial solution Perhaps I am using some of these terms, or your teacher is He may use some of these terms So it is good to know what we are talking about The typical polynomial form is just that Put the phrases in order of degree This is not a typical look If I want to set this polynomial as typical I will put this phrase first. So type 7x ^ 5 Then what is the smaller degree that follows? Well, we have the phrase x ^ 2 We do not have an expression x ^ 4 or x ^ 3 So it’s +1 – Okay, I don’t have to Type 1– + x ^ 2 Then I have this phrase, -5x Then my last phrase is -5 This is the typical polynomial form in which we have it In descending order of degrees Now let’s move on to solve a group of operations using polynomials This set of tools will be very useful later In algebra, or in the fields of mathematics Let’s simplify a set of polynomials We have been exposed to this in previous offers But I think it will give you a better meaning, in particular When we have phrases of even higher degrees So let’s say I want to add -2x ^ 2 + 4x – 12 And I’ll add that to 7x + x ^ 2 Now the important thing to remember when simplifying These polynomials are combining terms Containing the same degree of the variable I will quickly solve another example since I have several Variables present in the same case But anyway, I have these brackets here, but they are Actually, it doesn’t do anything If I had a quote here, I would have to Distribute the negative signal, but it does not exist now So I can write this as -2x ^ 2 + 4x – 12 + 7x + x ^ 2 Now let’s simplify it Let’s add phrases of the same degree And when I say the same degree, it should be It has the same variable as well But in this example, we only have the variable x So let’s do the addition Let’s see, I have this x ^ 2 phrase, and I have that x ^ 2 phrase So I can combine them I have a -2x ^ 2 – let me write them down First– -2x ^ 2 + x ^ 2 Then let me extract the x statements, which are 4x and 7x This is 4x + 7x Finally, I have this constant Here, E-12 And if I have -2 of something, I add 1 of The same, what do I get? -2 + 1 = -1x ^ 2 I can only type -x ^ 2 But I wanted to show you I am collecting -2 to 1 here Then I have 4x + 7x = 11x Then finally I have the constant, which is -12 And I end up with three limits Second degree polynomial The parameter here, that is, the parameter of the higher order term In the typical form – it’s actually the typical form – He-1 The coefficient here is 11 The constant is -12 Let us take another example from these examples I think you understand the general idea Now let me solve a complex example Let’s assume I have 2a ^ 2 b – 3ab ^ 2 + 5a ^ 2 b ^ 2 – 2a ^ 2 b ^ 2 + 4a ^ 2 b – 5b ^ 2 So here I have a negative sign, I have several variables But let us solve it step by step So the first thing we will do is Negative signal distribution So this first part we can write it as 2a ^ 2 b – 3ab ^ 2 + 5a ^ 2 b ^ 2 Then we distribute the negative signal, or multiply All of these limits are -1 because we have A negative sign outside So -2a ^ 2 b ^ 2 – 4a ^ 2 b And negative x negative = positive 5b ^ 2 And now we’ll collect similar phrases, so I have Phrase 2a ^ 2 b ^ 2 Are there any phrases containing a ^ 2 b ^ 2? Sorry, a ^ 2 b I have to be careful here Well, this is not ab ^ 2, no, a ^ 2 b ^ 2 Oh! I have here a ^ 2 b Let me write this down I have 2a ^ 2 b – 4a ^ 2 b These two phrases Let me turn to orange So here I have the phrase ab ^ 2 Now do I have other ab ^ 2 phrases? Other ab ^ 2 phrases No, there are no other ab ^ 2 phrases, so I will write them down -3ab ^ 2 Then let’s see, I have the phrase a ^ 2 b ^ 2 here Do I have any other one? Well, yes of course, the next phrase is Phrase a ^ 2 b ^ 2, so let me Write that + 5a ^ 2 b ^ 2 – 2a ^ 2 b ^ 2, correct? I wrote them Then finally I have the last phrase b ^ 2 + 5b ^ 2 Now I can collect them So the first group here is in purple, two elements of Something – 4 from this thing = -2 from That thing This equals -2a ^ 2 b Hence this phrase, they will not be combined with Anything, which is 3ab ^ 2 Then we can combine these two statements, if I have 5 elements from Something – 2 of this, I will have 3 elements From this thing + 3a ^ 2 b ^ 2 Then finally I have that last phrase, which is + 5b ^ 2 And we’re done We simplified this polynomial Here, by putting it in shape, you can think of it In different ways And the best way to think about it is Whole degree for the ferry Maybe we can put this first, but this one Back to you This is 3a ^ 2 b ^ 2 Then you can choose whether to put phrases a ^ 2 b or ab ^ 2 first. 2a ^ 2 b And then we have -3ab ^ 2 Then the phrase b ^ 2 here + 5b ^ 2 And we’re done We simplified this polynomial What I want now is to solve a bunch of examples of Create a polynomial In fact, the idea is to give you awareness Because of the importance of polynomials, with brief representations We will use it all the time, not just in algebra But later in calculus, and in everything So it is good things to see But what I want to do in these four examples is to act Area of \u200b\u200beach of these shapes using a polynomial I will try to coordinate the colors as possible So here, what is the space? Well, the blue part here, is its area Is x x y x x y Then what is the space here? It will be x x z So x x z But we have two of them! We have one x x z, and then we have Its last area x x z I can add x x z here Or I can type, + 2 x (x x z) And here we have a polynomial representing space The form is here Now let’s find the area of \u200b\u200bthe next shape What is the space here? Well, I have a x b ab This also appears to be a x b, so + ab This also appears to be ab, so + ab + ab I think it has been painted oddly Well, I will ignore these c Perhaps it is meant that this c Because that is the information we need Perhaps they are telling us that this is the rule, that is This, which is c Because that will help us But if we assume that this is also ab, which is what I will assume it until this offer Then we have another ab And we have this, which is a x c a x c This is the space of this shape It is clear that we can combine these four borders This is 4ab and then we have + ac It made this assumption and it contained a typo, where C had been given to us as an offer This box is here We don’t know if it’s square, and that’s only if a and c Equal Now let’s move on to this How do we find the area of \u200b\u200bthis pink shape? Well, we can take the whole rectangle, so that Be 2xy, then we subtract space These squares So each square has an area of \u200b\u200bx x x, i.e. x ^ 2 And we have two of these squares, then The area is – 2x ^ 2 Then finally, let’s move on to this This appears to be a dividing line So the area of \u200b\u200bthis point, of this area, is a x b, i.e. equal to ab Then the space here looks like ab as well So + ab And the space here is also ab So the total area is 3ab In any case, I hope this offer has been made clear to us Polynomials .

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