 Hey, hi friends. So in the last session, we saw the behavior of a quadratic polynomial on a graph and We also tried to tweak the values of the coefficients that is a b and c and we saw how the graph behaved or how the shape and Positioning of the graph changed on the graph paper. Isn't it now in this session? We are going to take it to the next level where we are going to discuss relationship between Between the zeroes and Coefficient of a quadratic polynomial. So, you know a quadratic polynomial px is defined as ax squared plus bx plus c where x is the independent variable a b and c are all real numbers and a cannot be zero and the reasoning is if a is zero it is reduced to a linear polynomial Now What are the zeroes zeroes are nothing but zeroes of a polynomial we saw in the previous session zero of a Polynomial, so what is zero of a polynomial? It's nothing but that value of x value of x Which makes px p of x as zero those are called the zeroes of a polynomial now. We also learned maximum maximum Number of maximum number of zeros zeros of of a quadratic of a quadratic polynomial maximum number of zeros of a quadratic polynomial is Equal to two right so a quadratic polynomial can have two zeros one zero or none zero no no zeroes at all Is it it we saw basis? The value of the discriminant Or we will be discussing it in further details later later on but graphically speaking if you see quadratic polynomial Can be of can be expressed in three ways or there are three varieties one is like this Yeah, where there is no intersection with the x-axis another will be just touching the x-axis and The and the third one is that it is intersecting at two locations Okay, so these are the three varieties of the quadratic and this is when the a is a is greater than zero So we we saw that the curve is opening upwards with a is greater than zero in a is less than zero Then again, there will be three varieties one is this Another one could be just touching the x-axis and third one could be intersecting the x-axis at two locations right so at max we see there are two roots here None here whatsoever in this one and there's exactly one here, right? So there are possibilities of one root no root and three or two roots sorry Now so let us say I should not be using the word roots. It's zeros. There could be two zeros One zero or none zero at all no zeroes at all right roots are used usually for equations now So what do we know we know then let us say let us say there are two roots of This quadratic polynomial and those roots are alpha and beta. Let's say alpha and beta are are roots of Roots of Px again. I'm using the wrong word roots. I should not be using roots. I should be using Zeros let us say zeros of Px Okay, now in in previous grades you might have understood learned this you know theorem above in algebra and that is if Px is a Polynomial it can be expressed as Gx times Qx times or plus Rx Okay, where Px Gx Qx Rx are all Polynomials where it has also given that degree of degree of Rx is less than Definitely greater than equal to zero and less than degree of degree of Gx and You must be aware. This is called the dividend Looks like similar right in arithmetic you would study dividend. This is my divisor This happens to be the quotient quotient and this is nothing but the remainder Okay, so the remainder cannot have a degree more than the divisor because if it is then you can go for further division Once again, so this is very important Now hence, let us say Gx is X minus some some Or in this case alpha itself, right? So let us say X minus alpha Okay, Gx is X minus alpha. So what will it become Px is equal to X minus alpha times some Qx some Qx plus Rx now clearly by this rule this rule What will be the? Degree of Rx. So X minus alpha is my Gx and its degree degree of X minus alpha if you see is equal to 1 It's a linear polynomial. So hence degree of Rx Rx will be nothing but 0. That means 0. It is a constant polynomial. Now by factor theorem, what do we know? We know that if X minus alpha is a R brother, we know that if alpha is is a 0 if alpha is a 0 of the polynomial Of the polynomial which polynomial polynomial Px If alpha is the 0 of polynomial Px then Then X minus alpha is a factor of Px Okay, if alpha is if 0 means if you put put alpha in place of X you'll get P alpha Is equal to alpha minus alpha and then Q alpha and then R alpha Right, but it is since alpha is a 0 of polynomial. This means this will be 0 If alpha is a 0 then P alpha will be 0. That means Anyways, this is 0 because of alpha minus alpha. So what is left is R of alpha is 0 That is the remainder itself is 0 So when the remainder itself is 0, that means clearly X minus alpha will divide Px right now when you Put let us say again once again if alpha is a 0 then if you put alpha in place of X Then R alpha R alpha happens to be 0 R alpha happens to be 0 means Reminder is 0 when remainder is 0 if you divide and what was the divisor divisor was X minus alpha So when you divided the polynomial Px by X minus alpha you got remainder as 0 That means what X minus alpha is a factor of Px. This is how It is denoted by a vertical line. So X minus alpha and is read as X minus alpha is A factor or it divides Px factor of Px Very good Similarly, if beta is also a factor then then X minus beta also divides Px, isn't it? Isn't it? So hence, what do we get? We get P of we can express now Px can be written as What X minus alpha times X minus beta times let us say Some another polynomial ux Right because X minus alpha is factor of Px X minus beta is a factor of Px so hence Hence Px will be let's say X minus alpha X minus beta times ux where ux is another polynomial Just to give you an example. Let us say if 2 divides 24, isn't it? Similarly 3 also divides 24. So hence 24 can be written as 2 into 3 into something else Let us say something else now that something is in this case 4 But clearly 24 can be expressed as one factor multiplied by another factor multiplied by something else Now if here if you see here, this is degree 1 Okay, and this is degree 2 This is also degree 1 that means the first two terms itself is a quadratic polynomial Why because if you multiply open this up and open these two factors You will get a term including x square isn't it when this x multiplies by this x You will get x square. That means this itself is a degree 2 polynomial that means the degree of ux must be 0 degree must be 0 because if it is not 0 then if it if it is even 1 Then the whole polynomial rhs the whole rhs will be of the degree more than 2 that is 3 4 like that right Which is not possible. Why because the lhs is of degree 2 So hence this something which was left over has to be a degree 0 term degree 0 term means it is a constant term constant term Let us say k. Okay, so what do I get I get px is equal to k times x minus alpha x minus Beta, so if you know the two zeros of the polynomial, you know the polynomial how Take any constant k and then do this calculation x minus alpha x minus beta example Let us take an example if 2 and 3 are The roots roots of px Okay Feel again. I should not be using the word root. I'm sorry for this. So again, it is zeros Are the zeros zeros of px Then the then px is nothing but if you see what any k whatever value you want to take as a real value k And this is x minus 2 times x minus 3 right another example another example if minus 3 and minus 1 are the roots again not roots sorry zeros of px then px is equal to x minus minus 3 Into x minus minus 1 that is x plus 3 times x plus 1 I hope you got it and then there will be a factor k as well k as well Right. So k could be anything so you can you can see it is 1 times x plus 3 times x plus 1 Or 3 times x plus 3 or x plus 1 Or root 2 times x plus 3 times x plus 1 Whatever value of k you want to take even your birthday you can take And this becomes the desired polynomial Now going further Whatever this px you got let us say you are expressing now by multiplying it or let us say you simplify this what will you get You'll get k times x square minus alpha x minus beta x plus alpha beta Okay, so this will become kx squared minus k times if you see alpha plus beta can be taken as common x plus k alpha Beta isn't it and if you remember what was our original px boss our original px was a x square Plus bx plus c if you don't Remember this is what I started with I started my discussion saying that px is a polynomial A x square plus bx plus c Okay So hence if you see there are two ways of expressing px one is this Right another way of expressing px is k times x minus alpha x minus beta And another way of expressing the same px is kx square minus k alpha plus beta x plus k alpha beta Now I can compare these two You know polynomials why because both are same so hence I can compare the coefficients and I can say a is equal to k right b is equal to minus k alpha plus beta Hence I can say b is equal to minus a alpha plus beta because k was a So hence alpha plus beta my friends is minus b upon a so hence I can say sum of Zeros of px px which is given as A x square plus bx plus c Is nothing but minus b upon a What is b coefficient here? What is a the coefficient here right which can be written as minus coefficient of I'm writing in short x divided by coefficient of x squared Okay, this is one of the results which we just Obtained very good fantastic Now if you compare the constant terms you will get c is equal to k alpha beta Is it it? If you check this is c this is c here. What do I get? So I will get alpha beta c upon k which is nothing but c upon a because a was k so hence I got product of product of zeros Of px Is equal to with px is equal to ax square plus px plus c Is nothing but c upon a which is nothing but coefficient of constant term or constant term divided by coefficient of x squared, isn't it? Now, isn't it beautiful that without even knowing the zeros without even finding the zeros you can find out the sum of the roots and product of the roots Is it it and hence you can also find the roots from this right? So, you know alpha plus beta is equal to minus b by a sum of the roots And alpha beta is equal to c by a product of the roots, right? This is one of the Findings and then hence px can be summarized as k times x square so hence if you see one of the expressions was this isn't it? So let me write that kx square Plus kx square then minus kx square minus kx square minus sum of sum of roots sum of roots times x plus product of product of Roots This is another way of writing The expression, isn't it? So let me take an example. So hence let us say if If Roots Sorry again zeros of a polynomial zeros of a polynomial polynomial polynomial px Are let's say one and minus two then then Find the polynomial find the Find the polynomial. How do I do that? Find the polynomial. How do I do that? So alpha is equal to one beta is equal to minus two So hence I can say alpha plus beta some of the roots is one minus two is equal to minus one product of roots product is equal to alpha beta is equal to minus two so hence px will be nothing but k times any k you can take x square minus alpha plus beta so minus alpha plus beta x plus alpha beta, isn't it? So hence it is k times x squared minus alpha plus beta is minus once it becomes x and alpha beta is minus two so minus two so this is a polynomial Okay, whose roots or sorry zeros are These alpha And beta so hence if the root if the zeros are given you can find out The polynomial hope you understood so just to summarize Please remember if px is equal to ax square plus bx plus c can be expressed as px equals to k times x minus alpha x minus beta where alpha and beta are the zeros Can also be expressed as a k times x square minus sum of sum of roots times x plus product of Roots please keep this In mind, okay, we will be solving some problems in the next session