 Hello and welcome to another problem-solving session on factor theorem and remainder theorem. In this particular question we are going to use remainder theorem to solve the particular question. Now the question is let r1 and r2 be the remainders and fx which is given by this polynomial xq plus 2x square minus 5 kx minus 7 and gx which is given by xq plus kx square minus 12x plus 6 r divided by x plus 1 and x minus 2 respectively. That means what x plus 1 is dividing fx and x minus 2 is dividing gx. If twice r1 plus r2 is 6, this is the condition. You have to find out the value of k. Where is k? So this k here and another k is here. So both are same, right? Okay, so let's try and solve this problem. Now by remainder theorem we know that if if fx is divided by fx is divided by x plus 1. Right? Then remainder is simply f of minus 1 y minus 1. If you remember what was our remainder theorem, if fx is divided by a x plus b okay, a x plus b is the linear polynomial, then remainder remainder is simply f of minus b by a. Is it right? So you can check here a is 1 and b is minus b is also 1 here. In this case, in the given condition x plus 1, given x plus 1, you will see a is also 1, b is also 1. So minus b by a is minus 1. So hence right, f of minus b by a, f minus 1 and how much will that be? So let's calculate f minus 1. It is minus 1 cubed plus 2 minus 1 squared minus 5 times k times minus 1 minus 7. Okay, so this is minus 1 and this one is 2 and this one is minus times minus plus 5 a minus 7. So net is 5 a minus 6. I think yeah minus 6. Right? It's not a, it's k. So 5k minus 6. Now this is for the first one. What? For the second one. For the second one remainder when g of x and which is given by that expression x cubed plus kx squared minus 12x plus 6 is divided by divided by what was the divisor in this case? x minus 2. Is it? x minus 2. Then remainder is how to find out remainder by remainder theorem. It will be simply f of not f in this case g because the dividend is gx. So g of again by the same logic in this second case in the second case you see a is 1 and b is minus 2 minus 2. Let me write it minus 2 Okay minus 2. So minus b by a is simply 2. So f or g of 2 rather in this case will give me the remainder. So that means 2 cubed plus k times 2 squared minus 12 times 2 plus 6 which is 8 plus 4k minus 24 plus 6 which is 4k minus 10 4k minus 10. Now what was given? So this is r1. Clearly this was the remainder 1 and this is r2 remainder 2 right? That's what they are saying r1 and r2 are the remainders respectively. So 2r1 plus r2 is 6. So it's given that 2r1 plus r2 is 6. That means what? What is 2r1 guys? This is 2r1. So this is r1. So 2r1 will be simply 2 times 5k minus 6 and r2 is 4k minus 10 see this one and this total is 6. So you have to find out the value of k. Now it's simple linear equation. You'll get 10k minus 12. Open the bracket 4k minus 10 equals 6. So 10k plus 4k how much? 14k and minus 12 minus 10 is minus 22 goes on the other side becomes plus 22 plus 6. So 14k is equal to 28. So k is clearly 28 upon 14. So this is the answer. k has to be equal to 2. What is the underlying concept which we used here? Use of remainder theorem to find out remainders without dividing. Okay, you could have gone for the longer division method and all that found out remainder and then put it into this equation and then find out the value of k. But since we know remainder theorem, there is no need of going for long division method and then finding out the remainders. This is why this is where we use remainder theorem.