Next application is almost end of the quadratic equation that is the theory of equations. So this is theory of equations. In theory of equations, we have We saw a relation between roots and their coefficients.
We saw in quadratic that sum of roots is given by minus b by a and product of the root is given by c by a. And that we know for the quadratic equation only. And we will see further that cubic equation, bi-quadratic equation or any polynomial equation of n degree.
So, what is the relation between its roots and coefficients? What is the relation between its roots and coefficients? Let's generalize Vimeo. So, we know the quadratic very well.
For ax2 plus bx plus c and its roots alpha plus beta that is given by minus b by and product of the root is given by c by. That's all the information we had. For cubic equation, for ax cube plus bx square, cx plus d equal to 0. If this is a cubic equation, then we have to divide it by a.
This is x cube plus b by ax square, c by ax plus d by a equal to 0. And if we know that if it is a cubic equation, it will have the three roots. Hot! 3 roots alpha beta gamma so alpha beta gamma so also this equation we can write like this x-alpha x-beta x-gamma equals to 0 now here this is an equation of 1 2 and 3 let's say we are discussing about the equation 2 and 3 so equation 2 and 3 are same equation 2 and 3 are Same, and we have to do the same thing in that too. If we check the coefficient of x2 of x3, then the coefficient will be minus b by a only, and in second plus, then it will be minus.
I am not expanding it here. Now if we expand it fully, then x3's coefficient, then x2's coefficient, x's coefficient and constant term. if we check their coefficients and finally these coefficients will be in the form of sum of roots or will be in the form of some of the pairs of roots alpha beta plus beta gamma plus gamma alpha and their product will be in the last so that is alpha beta gamma so for this we will have if alpha beta gamma be roots then you Jiga alpha plus beta plus gamma and that is given by minus B by a and so go up you have to write the sum as a product of pairs sum as a product of pairs means that is alpha beta beta gamma plus gamma alpha you value miliki yeah minus B by a So, we are going towards generalization.
So, we can write it as minus 1 to the power 1 b by a. And we can write it as, here the sum of two roots is minus 1 to the power 2 c by a. So, we want to catch the generalization pattern.
That's why we are writing it like this. and third one will be product of three alpha, beta, gamma so that can be the d by a and here also I write minus 1 to the power 3 because this is product of three roots so we will put minus 1 to the power 3 and that value is d by a and this value will also be minus d by a but now we want to catch a generalization number of these two and a generalization number of this Similarly, if you have a biquadratic equation for ax4, bx3, cx2, cx2 plus dx plus bx3, plus E now this is dx plus E equal to 0 if alpha beta gamma Delta B roots then And in this also, the same point will be given that the sum of these two will be like this. In sum, we will write alpha plus beta plus gamma plus delta and that is the minus b by a. By the way, to remember in short, there is a method that we put minus here, then plus, then minus and plus.
And we will divide the coefficient by a. So you can write the x4 plus b by a x cube plus c by a x square plus d by a x plus e by a equal to 0. If we see the equation in this way, it will be better. And with this minus, with this plus, then minus, then plus. and it works in the same way and this value will be minus b by a you can write it in the same way minus 1 to the power 1 b by a then we saw here that the sum of individual three roots In years, there was a sum of 2 products and then there were 3 products.
Exactly this thing is moving forward in generalization. In this also we are going to get the same thing. Alpha, beta. So in this, suppose out of 4. select 2 numbers 4C2 is 4x3 1x2 so these are the 6 total values not writing the whole 6 values but writing summation of alpha beta so from this pattern out of 4 pairs 4C2 you can find out the total 6 pairs are there six different pairs are there of the roots so this value will be minus b by a and c by a then see this is the product of three so we write it like this alpha beta gamma and here is the summation sign the meaning of summation sign is that in this way all the possibilities will be created we took three out of four so, if we select 3 from 4, then 4C3 will be there which is equal to 4C1 and in this way, the triplets of 3 3 in the form of product from 4, the product of 3 3 will be taken so, that will be the total 4 so, we will have the total 4 cases but, we don't have to write all we will write it and leave it and the 3 will be there after this minus d by e and then after this product of 4 will come so product can be written as alpha beta gamma delta directly and that value is equal to here minus is applied and then plus will come this will be e by e now see the generalization we have taken here see that generalization here here it was adding single individual roots so power was 1 If I write this as c by a whole square and write it as minus 1 whole cube d by a then there is no problem. And write it as minus 1 to the power 4 d by a.
So you understood one sequence. If product of 4 is running here, then here minus 1 to the power 4 and write it like this. If product of 3 is running, then minus 1 to the power 3 d by a.
Let's move on to next. So, in this way, we will talk about general equation, general equation, polynomial equation of degree n. If there is a polynomial equation, let a0xn plus a1xn-1 plus a2xn-2, a3xn-3 up to an equals to 0. Equation of degree n.
Equation of degree n. Let this as a n degree equation. Where a0 is not equal to 0. And you can say a0. a1, a2 upto an belongs to real number.
These all are real numbers. This is the polynomial equation which we defined in the starting. But if degree is n, then it has n roots.
And let those roots be alpha1, alpha2, alpha3, alpha4, alpha5. up to alpha n b n roots suppose it has n roots like this so initially we have seen one thing that first individually one root sum comes then in pairs their two pairs sum comes then triplets sum comes then 4 sum comes then 5 sum comes in the end all product comes so we will work out on it like this and we have understood the generalization of the sum of roots so we will write the same therefore this is known as the sum of roots we will not be able to write all of them we don't know the n degree so for that we have to work out the generalizing value in general way so you will write the sum of alpha i and i will be from 1 to n It means that all the roots are summed alpha1 plus alpha2 plus alpha3 up to alpha n and here I told you that you will write the coefficient of this so better in general form write like this write power 1 of minus 1 and write a1 by a0 this is best now the second thing we are going to write is we have to take products of alpha i22 so from these roots we have to select the 2 at a time out of n so basically nc2 total roots will be made total pairs will be made like this nc2 total pairs will come but we can't write that in general way how much will come so there is a way of that write alpha e alpha j and i is not equal to j it will take start from 1 and will go up to n you will write its value here we have learnt minus 1 to the power 2 and next write a2 by a0 so this is a2 by a0 similarly if you write 3 then write alpha i alpha j and alpha k where you write i is not equal to j is not equal to k equal to 1 means any number will start from 1 alpha 1 will be there then alpha 2 and this alpha 3 will have to be there but it cannot be equal to 1 So, out of n roots, the triplets of 3 will be different from each other. It cannot be repeated. So, the condition which we have written like this, there will be no repetition.
And that goes up to n and this value will be equal to minus 1 to the power 3 because it is of 3 and a2 by a0 is there. After this, a3 by a0 will come. So, In this way, similarly, if the polynomial is of 3, then 4, 5, and if the polynomial is of degree 11, then the product of the last 11 roots will be alpha1, alpha2, alpha3, up to alpha11.
So, this is a way of writing it. It has a greater pi sign. and its value is i equals to 1 to n and here you have to write alpha i this basically indicates alpha 1, alpha 2, alpha 3 up to alpha n product and this value is equal to minus 1 to the power n this is the generalization we have learnt in degree 3, degree 4 and after that when we went to n degree so here this is the product of n roots it means you have to write the value n here and that is the last number constant term so this is the plus here last a n upon a 0 so you have to write down the a n by a 0 and this is the best to find out so we suppose what will be the product of all its roots so the product will be power 17 so you can see the constant term divided by x to the power 17 we will divide it by x to the power 17 minus 1 to the power 17 here we can see an example if we ask this kind of example this is 2x to the power 17 plus 19x to the power 3 and minus 11 suppose I am writing a very easy question and the highest degree is 17 and it will have 17 roots and the question is what will be the product of roots of this quadratic find out the product of roots of this polynomial equation of degree 17 so that will be your product you will write the total product here we have derived the formula it will be minus 1 to the power 17 and we will write the coefficients minus 11 by 3 2 right so this is negative so overall it will give the positive value so they give all fast up with me workout curse at their directly I'm gonna call the key is your degree 17 key polynomial equation a polynomial equation of the degree 17 it has the 17 roots so this is the product of all the 17 roots so there are some applications which can be used here and directly we can work out that value after this we will take some examples after this we will take some examples in theory of equation The conditions we have understood, let's assume that if alpha, beta, gamma are roots of cubic equation x cube plus qx plus r equals to 0, then find Question a question whose Roots are whose roots are this question get in parts and suppose alpha plus beta beta plus gamma and gamma plus alpha Anagama, Libby a key would question Marani a just a roots alpha plus beta beta plus gamma or gamma plus alpha So we have to create that equation.
Second is alpha beta beta gamma gamma alpha and third is alpha square beta square gamma square. See, I have told you the concept of transformation of equation. So we will use it.
You can use it wherever you want. Because there is no symmetric change in it. This will definitely be used in this and let's see how we will work out and he has given the alpha beta gamma are the roots of this ad cubic equation So first you will directly write sum of roots alpha plus beta plus gamma that is minus b by a that is the coefficient of x square and when the term x square is absent it means its value will be zero So from here we are getting a hint for the first one we need sum of roots so we can write alpha plus beta equals to minus gamma and beta plus gamma which is second root beta plus gamma equals to beta plus gamma equals to minus alpha and third is gamma plus alpha gamma plus alpha equals to minus beta so see it is interesting these are the three roots and this The roots of the equation are alpha, beta and gamma What changes have been made in the other three roots?
The changes have been made in the three roots that they have been multiplied by minus And otherwise all the three roots are same This is minus alpha, minus beta and minus gamma All three roots are there but they are multiplied with a minus sign So for that, the transformation we saw in quadratic is same Transformation will be used here also. So change x by minus x to get required equation. To get the required equation. We will do that. We will change x to minus x.
Minus x whole cube plus q into minus x plus r equals to 0. So this is minus x cube minus qx. plus r equal to zero and to set it properly multiply it by minus x cube plus Now see this is the required equation. This is the equation whose roots are alpha plus beta, beta plus gamma and gamma plus alpha.
Because it said that roots are like this. And when we wrote the values and used the given result, we were seeing that all three roots came out. Alpha, beta and gamma but it is multiplied with a minus sign. So we understood the same point in transformation.
We have multiplied it by the minus sign. So replace x with minus x and we got the required equation. Whose roots are alpha plus beta, beta plus gamma and gamma plus alpha. After this, let's take next.
See, it is showing its roots. Alpha beta. beta gamma gamma alpha second question over accuracy this moment see question Manani a diskey roots alpha beta beta gamma or gamma alpha so be a paper office product of the root key back correct for given equation for given equation alpha beta gamma Product of the root, the last point is that in general we write in cubic.
ax cube plus bx square plus cx plus b equal to 0 sum of root minus b by a alpha beta plus beta gamma plus gamma alpha q value c by a then minus d by a so here the coefficient here also the constant term r will take minus minus r will come minus r by 1 so we can write like this right so we will create something alpha beta value minus R by gamma. Let me write it clearly. This is gamma and this is R.
Right. This is R and gamma. In beta gamma, this will be minus R by alpha beta.
This is beta gamma. Beta gamma will be minus R by alpha. Alpha beta beta gamma gamma alpha.
Gamma alpha will be minus r by beta. Right. Now we have to do one thing on this. That the values of these three roots are coming in this pattern. What are those roots doing?
The reciprocal of all three roots is happening. and is multiplied by minus r all three roots are multiplied by minus r and their reciprocal is happening so you can do like this we will let it we will work out like this let minus I will talk about this. Minus r by alpha. We will let minus r by alpha be y. If we let this be y, then what will be the value of alpha?
So alpha will be equal to this will go up and that will go down. Alpha equals to minus r upon y. And what was alpha?
It was a root of given cubic equation. Given cubic equation was a root. Means it was a value of x. It was a value of x. So we will do this.
We will replace alpha value of x in this. So therefore, required. equation required equation can be minus R upon Y whole cube plus Q into minus R upon Y plus R equal to 0 so this is minus R cube upon Y cube minus minus qr upon y plus r equal to 0 multiply by y cube minus r cube will come multiply by y cube minus qr y square and plus R y cube R y cube value will be equal to 0 ok this equation is almost done let's do one thing let's reset it write R y cube minus Q R y square minus r cube equal to zero is required equation is required equation now see here it does not matter we have taken it out in y variable whereas original equation was given as x whatever the variable is we can write it as t we can write it as any other variable so If you want to reset it, change the form in which it was given and then change y to x It can be written as rx cube minus qr x square minus r cube equal to 0 Means I have changed the y variable by the again x variable Now it can be t variable or it can be any variable The QBQ equation is the same.
Okay. After this, the third example comes. See, it has squared the roots. So, I told you about the condition in transformation. If it will square, then we will do the opposite.
X we will change it to root x so replacing for third part we will use transformation directly we will use transformation as we used to do in quadratic because all three roots have symmetric change roots are alpha,beta,gamma and it is square so we will do one thing we will replace it in transformation Replace x by root x or x to the power half x to the power half Replace x to the power half in this given equation So this is x to the power 1 by 2 2 to the power 3 plus q x to the power half plus r equal to 0 Now see the requirement is that this is not its general format It is not a cubic equation because This value is 3 by 2 plus q x to the power half. What we will do is reset the whole equation. We will set it in cubic format.
Cubic format means general cubic equation. We will have to write it in the same way. So we have to square both the sides. Squaring both the sides. It will be x cube.
And plus q square x. and plus 2ab term is 2ab and x to the power 3 by 2 plus half power will be 2 equals to r square or you can write the x cube this is x cube we will take it plus 2qx square and plus q square x minus r square equal to 0 so this will be the required equation simply by replacing x by root x and then converting into the general format of the cubic equation gives you the required equation whose roots are alpha square beta square and gamma square. Let's take one more example find relation relation between between PQR PQ and R if if roots of the fruits of the cubic equation cubic equation x cube minus px square plus qx minus r equal to 0 are such that they are in AP. the roots are in arithmetic progression and the condition is given and you have to find out the relation you have to find out the relation between the pqr and for this cubic equation and he has given the condition that roots are in ap if roots are in ap and we will let the roots be alpha beta gamma so as per given condition to be tight was to alpha plus gamma because they are in AP middle term car double don't know extreme terms k some cake Paloma See, and then we do one more thing, we add 1 beta to both sides. So, this will be equal to 3 beta equal to alpha plus beta plus gamma.
And see one thing here, 3 beta equal to, sum of root is written for this equation minus b by a. So, its value is simply p. Its value is p minus b by a.
So, it will be minus minus plus. So its value is p. So can we write the beta equals to p by 3?
Beta will be the value of p by 3. Beta is the root of this cubic equation. It will satisfy it. So we can place the root value of this equation in the equation.
Because its root is p by 3. It is the root of this equation. So putting x equals to p by 3 in equation. x means x is the root. x becomes the root of any equation.
So in place of x you have to put the root of the equation. So this will be placed. This is p by 3 whole cube minus p into p by 3 whole square and plus q into p by 3 and minus r equal to 0. So this is the P cube by 27 minus P cube will be made. Total P cube will be made but below 9 will come. Plus P cube upon 3 minus R equal to 0. And if I want to solve the values of P cube, we will multiply it by 3. Or we will take 27 LCM in it.
Or you multiply it by the 27. That will be better. multiplied by the 27 overall so it gives the p cube minus 3 p cube and plus 9 p cube minus 27 r equal to 0 this can be reset this is minus 2 p cube plus 9 p cube minus 27 r equal to 0 If you want you can multiply it by minus. We want to write the condition in correct format.
So we multiply it by minus. 2p cube minus 9p cube plus 27r equal to 0. This is the relation we wanted. We wanted relation between p and qr. The roots of this quadratic equation are sorry cubic k they are in ap that is in ap so roots were in ap we wrote its condition from here we got to keep value directly and from that beta came out beta which is the root we placed it in equation and simplified it and we got the relation between pq and r after this we are going to use the last application of quadratic equation that is the Descartes rule to decide how many positive roots are there how many negative roots are there the root should be real.
Descartes rule so our next application is Descartes rule to check how many roots have the positive roots and how many roots are negative to check the sign of the roots only sign gives information not the roots only tells how many are positive roots and how many are negative roots