 Hello friends, welcome to another session on problem solving on factor theorem and remainder theorem Now in this question, it's given that if x square minus 1 now, please note It's x squared minus 1 is a factor of a x square a x to the power 4 Plus b x cubed plus c x squared plus dx plus e We have to show that a plus c plus e is equal to b plus d is equal to 0 Okay, so in such case questions how to proceed so it's it's given so what is given given is x square minus 1 is a factor of This particular polynomial. What does this mean? It means that if you divide the given polynomial by x square minus 1 You will get remainder 0. Is it so hence? Can I not say that for? some so right me let me write it out solution for some polynomial For some polynomial Gx let's say we can say FX is equal to x square minus 1 times gx correct What does this mean? Let us explain. What is FX first of all? So FX? I am saying is our given polynomial which is a x to power 4 b x cubed plus c x squared plus dx Plus e It said or it is you know given that x square minus 1 divides this FX completely What does it mean x square minus 1? You know when you divide FX by x square minus 1 you'll get remainder 0 That means x square minus 1 times some other polynomial will lead to FX if you remember our basic equation of dividend is equal to what? divisor Into quotient Right for polynomials also it holds we have seen in the previous sessions Dividend is equal to divisor into quotient plus remainder. This is the underlying Principal now remainder here is Zero why because x square divisor is a factor, right? So in this case if divisor is the factor then remainder has to be zero This is what we have learned, right? So if remainder is zero only then we can say that this will be a factor of dividend We have learned this in the previous Sections or sessions right so 20 is equal to five times four Plus zero right when you divide 20 By five which is the divider the remainder has to be zero only then we can say five is factor of 20 Just an example to illustrate this fact now FX is equal to a x to the power 4 b x cube plus c x square plus dx plus e has a factor x square minus 1 So I can write a x to the power 4. Let me use the same ink So here I can write a x to the power 4 plus b x cubed plus c x squared plus dx plus e Is equal to x square minus 1 I can write this as x minus 1 x plus 1 and for some Polynomial gx which I am not interested in finding the you know exact Polynomial right so some gx so some gx is a polynomial which when multiplied by These x square minus one which is which itself can be factored into x minus 1 and x plus 1 Will give you this particular polynomial. I hope this is understood by all right So once again, I'll try to explain once again. So this is a polynomial and X minus 1 x plus 1 that is x square minus 1 this entire thing is x square minus 1 is a factor of LHS when can that be possible when you have some other polynomial gx which when multiplied by x square minus 1 Will give you This that's what is the basic definition of factor. There is no remainder. The remainder is 0 Okay, now if that is so this is the given relationship now This is where we have to or this is what we have to use to find out The given, you know statement or whatever has to be proved here a plus c plus c is equal to b plus d is equal to 0 now What happens if I put x equals to 1? If x equals to 1 let's put x equals to 1 so what will happen everywhere wherever you see x put 1 So this is a 1 to the power 4 plus b 1 to the power 3 plus c 1 squared plus d times 1 plus e is equal to 1 minus 1 and 1 plus 1 into g of 1 Okay, why did I choose 1 in the first place? This is simply because if I choose take 1 this right hand side becomes 0. Is it 1 minus 1 is 0? So the moment I put x equals to 1 I get a relationship between a plus b plus c plus d plus e and that is a Plus b plus c plus d plus e is 0 First equation or first relation. I got simply by putting this x equal to 1 The RHS became 0 and I can put the same value of x in the left-hand side as well I got this relationship. I would have guessed by now. What is the you know the next step? I'm going to take I'm going to say x is equal to minus 1 again if you see if x equals to minus 1 The right hand side is going to be 0 once again and what will be the left-hand side a minus 1 to power 4 plus b Minus 1 to power 3 plus c minus 1 to power 2 plus d minus 1 Plus e and this is again 1 and Then minus minus 1 times 1 plus minus 1 into G 1 Minus 1 sorry g minus 1 see it doesn't matter. What is the value of g 1 and g minus 1? Why because anyways this is going to be reduced to 0 because of this now here also this is going to be 0 So if this is 0 again LHS becomes 0. So hence a this will be now minus b Plus c minus d plus e is 0. So this is 1 and This is 2 Correct so from here. What do you get from here itself? You'll get a Plus c Plus e is equal to b plus d correct a plus c plus e is equal to b plus d Which is the second part of the given Condition to be proven. So I'm sorry first part. So we have proved this we have to now prove that both of them Is are equal to 0 so a plus e plus c is equal to b plus d now from 1 What have we got a plus b plus c plus d plus e is 0 now Club them like this a plus c plus e Plus b plus d is 0 Now from 3 We can say that b plus d is equal to a plus c plus e. So this a plus c plus e I Can replace very well by b plus d. Isn't it from? 3 plus this b plus d is anyways there this one This is equal to 0 so hence twice b plus d is 0 That means b plus d is simply 0 and since b plus d was equal to a plus C plus e from 2 or from 3 From 3 we can say B plus d is equal to a plus c plus e equals 0 This is what we have to prove you have to prove correct So what did we learn? We didn't apply any factor theorem or in fact we did apply factor theorem That was the essence of factor theorem the way we prove factor theorem So factor theorem is nothing but if you if you know x minus 1 was supposed to be the factor of this So f of 1 will be 0 similarly if x plus 1 is a factor of LHS polynomial analysis then f of minus 1 will also be 0 this is what we Deployed over here But then we never you know took the name of factor theorem because the essence is this only this particular behavior that dividend Is equal to divisor into quotient plus remainder and when remainder is 0 then divisor and quotient both become The factor of dividend and that's what we have utilized over here then deployed the value of x suitably so that we get the required Relationship right you can you could have used You could have used the factor theorem also saying that if x square minus 1 is factor of x fx then both x minus 1 and x plus 1 would be factors of fx Then apply factor theorem and you will get the same result I Hope you understood this proof