Polynomial Theorems | Class #05 | A2 Mathematics 9709

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Okay, so in the last class we discussed it was the first class for polynomials and we discussed what is a polynomial and we also learned how to do the long division, how to divide a polynomial by a polynomial and how do you write it at the end. And we also discussed that if the after division if the remainder is zero this tells us that the divisor the thing which you are dividing it is a factor. A factor completely divides another thing. When it when something completely divides another thing the remainder there is no remainder. So if you are getting a remainder this means that your divisor is not a factor. And if your remainder

is zero this tells us that the divisor is a factor. So we will use this logic later properly again as well. But right now or to strengthen the concept of the long division we have two equations over here. Sorry, two questions over here. Or bracket. So they have given you two they want you to divide x power 4 x power 4 + 3x cubed + x squared - 2 by a quadratic factor or divisor. x squared + x + 1 So the logic will be the same. First of all you choose the uh first term of your divisor x squared and you will see you will see that you will have to multiply x squared with x squared so that it becomes x power 4.

So this becomes x power 4 and then you will multiply the rest of them with the x squared. So this would become plus x cubed and then plus x squared. And then you change the signs for all of them. So this becomes minus, this becomes minus, this becomes minus. And then you add them. So, x squared minus x squared uh sorry, x power four minus x power four cancels out. This becomes 2x cubed. And this also cancels out. x squared minus x squared. So, this also cancels out. And minus two would be carried forward. And now you will multiply the x squared with 2x only. So, that you will get 2x cubed. So, this becomes

2x cubed plus now you multiply the x with the 2x. So, 2x squared and then one with 2x. So, plus 2x. And then you again change the signs for each of the terms. So, this is minus now, this is minus, this is minus. And then this cancels out. Minus 2x squared carried forward drop it down. Minus 2x squared minus 2x and minus two. And now you see that you just have to multiply the x squared with minus two. You multiply x squared with minus two, you will get minus 2x squared. And then you will get minus 2x and then minus two. Again change the signs. >> So, zero I get up cuz that means it's a factor. >> Zero zero zero. So, the remainder is

coming out as zero. So, you will say that this was a quadratic factor. >> Mhm. >> It's a quadratic factor therefore. >> Take over there. >> Okay. Then we have another equation. Sorry, another question. Also similar. They're asking you to find the quotient and the remainder when f x is divided by They've given you f of x. This x cubed plus 3x squared plus 4x plus two and this thing is divided by x squared plus x minus one. So, you will multiply the x squared with only x. >> x >> Yes, I didn't change the sign. Then you add them to zero. >> 2x squared and then 5x >> Then carry forward the plus two.

Now, you will multiply with >> two >> So, 2x squared >> plus 2x minus two. So, yeah. Plus two, yeah. Two minus two, yeah. That means that 3x plus four >> And that is it. So, that you will write it down. The quotient is the part of x plus two >> Quotient is x plus two and remainder is 3x plus four. >> 3x plus This is it. Now, the second part use the factor theorem to show that x plus one is a factor. Now, this is something which is the new. We don't know about this. This is the new topic. So, what is this one? Now, you know how to do the long division. Why is long division good? Long division can help you

help you check if it is a factor or not. >> Mhm. It can help you find the remainder. And it can also help you to find the quotient. So, this is the complete package. It will help you Uh if you only know this thing also, you can solve the all of the questions. Almost all of the questions. 95% of the questions are possible. But, there's another one more thing. That is uh more uh that is included, which is the remainder factor theorem. What is this? Uh this the theorem tells us that if you have a function if you have a polynomial f x. Now, this can be any polynomial. Cubic, anything. If it is a polynomial

anything. And they have given you a divisor. They are asking you to divide. Divisor is x minus a. So, this would tell this would mean that f a if you can put the put the value of from the divisor into the function, this will get you the remainder. Directly. And if f a is equals to zero, means the remainder is zero. And so, therefore, the x minus a is a factor. Now, how do we do this? So, if you take the same question let's take the same question. They to check if x + 1 is a factor or not. So, the divisor is x + 1. And the function is the polynomial is this thing. So, if I write it down, f x is given to us

x cubed plus 3 x squared plus 4 x plus 2. And then look if the divisor is x plus 1, what would you do? You first of all find the value of x. So, x + 1 is equals to 0. You solve it for the divisor value of x and then x will come out as minus 1. And now if you input the value of minus 1 over here, the value from the divisor into the question again. And then use your calculator. See what do you get? So, you would get minus 1 3 minus 4 and plus 2. So, how much would this be? Zero. I think zero. Minus 5 plus 5, so zero. >> Mhm, zero. >> telling us that the remainder is zero. And if the remainder is zero

this means that x plus 1 is a factor. So, the remainder factor theorem can be done directly without any long division or without any critical working you can you can directly get you the remainder for whatever the uh divisor is. So, the benefit over here is remainder factor theorem this is quicker and shorter and less working. Gets you the remainder. But the problem is it will not get you the quotient. After dividing something, it will not give you the quotient. Quotient, if you if you need the quotient, you will have to do the long division. Well, this this part is the quotient. The remainder factor theorem can only

help you work with the remainder, remainder part. If it has remainder zero, then it is a factor. If it is not zero, then it is not a factor. But this is more quicker, more shorter working, less working. So, depend on the question. Right now, we could have also did we could have also done the long division. With this uh divided by uh three, this divided by this, and the remainder after completing would come out as zero. That would also tell us that this divisor is a factor. But the problem is they are asking us to use the remainder factor theorem, the factor theorem. So, this was the factor theorem. So, you get the value of x from

the divisor, and put it back into the function, and find the value of the function. And the value of the function is that remainder. And if the remainder is zero, then factor. Okay? But they can ask you to fully factorize something. Let's see. Okay, here it is. So, they have given you a polynomial in which there is an unknown A. We don't know A. And it is given that 3x + 1 is a factor. Is a factor. Now, we know that when something is a factor, the remainder will be zero. So, over here for to find the value of A, we can do long division. As well, we can do the remainder factor theorem as well to find the value of A.

But, the better the shorter method is the definitely the remainder factor theorem. Because, you know what is the remainder. So, if your function is this, let's suppose f x is equals to a x cubed - 20 x squared + x + 3. So, if I find the value of x from this divisor, you will get x is equal to or 3 x + 1 is equal to zero, and so x will be equal to -1 / 3. So, you will say that if you find the value of the function at 1 / 3, -1 / 3, it will come out as zero. Because, it is a factor. So, your equation becomes zero is equal to A -1 / 3 cubed plus Sorry, minus 20 -1 / 3 to the square plus -1 / 3 and + 3.

And then, you can use a calculator. -20 -1 / whole square, and then minus -1 / 3. The negative and then, you have plus plus three as well. 4 / 9. So, this will be equal to zero. -A / 27 and + 4 / 9. So, A / 27 is equal to 4 upon 9. And then therefore, A would be equal to 4 multiplied by 27 upon 9. is equal to 12. So, the value of A is coming out there. Well, that's it. Do you get it? >> He says >> Yes. Yes, yes. And then when A has this value, factorize. So, this means now the equation has become 12 X cubed - 20 X squared + X + 3. 12 X cubed - 20 X squared + X + 3. And then uh now you have to factorize. So, this is a

cubic equation. Whenever you have a cubic equation, you have three factors. A quadratic equation has two factors. A cubic equation has three factors. And if it is a quartic equation, you will have four factors. So, you already have one factor, 3 X + 1. This is the first factor they've already given to you. Now, we need the other two factors. And to find the other two factors, you would need the quotient. But the long this long division Sorry, what do you say? Remainder factor theorem does not give you the quotient. So the remainder factor theorem is only helping you in the first part. So to get the quotient, you will have to

do the long division. Of the equation that we you have you have the curve. And now you will divide it with the factor, the first factor that you have, 3x + 1. And this would help you to find the quotient, which will have the other two factors which you want to find. So this this would be multiplied with 4x squared. So I will quickly do this, 4x cubed plus 4x squared. Change the signs. Add them. -24x squared plus x. And then minus eight x. So -24x squared -8x. Change the sign. Add them. 9x + 3. So plus three. So 9x + 3. Change the signs. Add, you will get zero. So look, your remainder is zero and this

is the this quadratic quotient. If you factorize it further, you can factorize it using middle term breaking. You can do 6 * 2. You factorize this would become 4x squared -8 -6x -2x +3. So 2x common x - 3 and minus one common x 2x - 3. So 2x - 1 and 2x plus minus three. I think you look like I made a mistake. Okay. Okay, yes, yes. This means now that the three factors are what? You found the first factor that they had already given you given that to you. It was 3x + 1. And the other two factors you got after factorizing the quotient. The 2x - 1 and 2x - 3. Okay, this is the answer to your third

part. Long division to the quotient I had the quotient after the long division. That comprises of the other two factors. We got the other two linear factors of You wanted the three linear factors one they had already given me two and to find other factors you needed the quotient and to find the quotient you did the long division. And after long division you got the quotient and you factorized the quotient you got the other two. Okay, these are also very uh generic types of again repetitive questions. You will see a lot of questions like these. The polynomial ax cubed plus bx squared plus 5x minus 2

is denoted by px. So, there are you can see two unknowns which you want to find. A and B. And it is given to that given you that 2x - 1 is factor. And it is given that x - 2 gives the remainder 12. So this means that using these two conditions you can form two equations. And then using the two equations you will solve them simultaneously and get the answers to the two unknowns. So if you have 2x - 1 is equal to zero, so your x will come out as 0.5. And because it is a factor, this means that function P and if the input is 0.5, the resultant would be zero. Because it is a factor. And if x - 2 is equal to zero,

so x is equal to two. So if your function inside your function you input the value of x as two, you will get the remainder as 12. Because they've given the remainder. So you will form these two equations, simplify them, form quadratic simultaneous equations, solve them. So this would become zero is equal to 0.5 cube x sorry, A plus 0.5 square B plus 0.5 into five minus two. And you can use your calculator to simplify these things. 0.125 A 0.25 B and this would become 2.5 minus 2 0.5. So -.5 = 0.125a + 0.25b. This is your first equation. Now, you will use P2 and it is 12. Uh = 12. And if your x is 2, so 2 ^ 3 a

+ 2 ^ 2 b + 5 * 2 - 2. So, this will become 8a + 4b + 10 - 2 = 12. So, 12 - 10 + 2. So, 4. 8a + 4b. And now you will solve them simultaneously. You can use your calculator to solve them simultaneously. So, 0.125 0.25 and -0.5. Sorry. Then 8 4 4. So, your value of a a is coming out as 2 and b is -3. That's it. Is your answer. And you can input these values back again. Then they're asking you, when a and b have these values, find the quadratic factor of px. This becomes now Now, your function px has become Uh a is 2. 2x^3 - 3x^2 + 5x - 2. And we already have one factor. 2x - 3 was the factor. And they're

asking you to get the quadratic factor. So, you need the quotient, to get the quadratic factor. So, you need the quotient, right? Your your long remainder factor theorem doesn't give you the quotient. If you want to find other factors of a particular polynomial, you will need to do the long division. Mhm. 2x³ - 3x² + 5x - 2 and 2x - 1. We will multiply it with x squared. So, 2x³ - x² change signs. Cancels out. - 2x² + 5x. We will multiply this thing with - 2 or no, - x. So, - 2x² and + x. Again, change signs. Cancels out. 4x - 2 and this would now be multiplied with + 2. So, 4x - 2. Change the sign.

So, 0 0. So, your remainder look, it is coming out because you're dividing it with the factor, the remainder should come out as 0. And this is the answer to your last part. This is your quadratic factor. The other quadratic factor. Okay? Any questions?