 Hello and welcome to another session on problem solving related to factor theorem in this question It's given that if a y cube plus b y square plus y minus 6 has y plus 2 as a factor So y plus 2 is a factor of this given polynomial in y and leaves a remainder for When divided by y minus 2 find the value of a and b This is the question now This is a mix of both remainder as well as factor theorem. It seems both the theorems are going to be utilized over here. So let's Utilize the knowledge of our factor theorem and remember theorem to solve this question now, so Let us say Fy Right. It is a polynomial in y. So let's say f y is a y cube Plus b y square Plus y minus 6 Right. This is the polynomial in y now given that y plus 2 is a Factor of f y and Then hence, what do we know then f of minus 2 must be 0 If you recall what was factor theorem, let me Write it here, right? So if f x or rather the factor theorem suggested said that if if a x plus b is a factor of Fx Then what do we what did we have then f of minus b upon a is 0 Right. If you compare in this case a x plus b is y plus 2 Right here clearly The and now don't get confused between this a here This a this b is Different from this a and b Right. So this is a general generic term a x plus b in this question We are supposed to find this a and this b. So don't get confused Okay, so general a x plus b if it is a factor of f x then f of minus b by a is 0 In this case our a x plus b is y plus 2 Only the variable x is replaced by y value of a that is coefficient of the variable is 1 and The constant term is 2. So hence minus b by a is minus 2 Correct. So and see if you see here. I'm using f of minus 2 is 0. So let's deploy a f minus 2 So a times minus 2. So wherever you see why write minus 2, correct? This is what is meant by value of the polynomial at y is equal to 2 or minus 2 rather in this case Minus 2 minus 6. This must be 0 Is it so let's solve minus 2 cube is minus 8. So minus 8 a minus 2 square is plus 4 b and Then minus 2 minus 6 is minus 8 is equal to 0 Is it so you got the first equation you can always simplify it and this will be you can Take away 4 from or minus 4 rather. So if you divide this entire equation by minus 4 What will you get you will get 2 a minus b? Plus 2 is 0 Right, so this is equation number one Then by the second condition. What is the second condition that it leaves? remainder 4 when divided by y minus 2 so then why remainder theorem by remainder theorem, you know that If fx is divided by ax plus b then f of minus b by a is the remainder. So in this case f of 2 Will be equal to 4 Is it right? What is the remainder theorem guys? The remainder theorem says if fx is divided by Any linear expression ax plus b ax plus b then remainder is simply F of minus b by a Right remainder is equal to minus b by a right in factor theorem. This is the same remainder becomes 0 now Reminder is given to be equal to 4. So f of 2 must be 4 So f of 2 let's kind of let's find out f of 2. So by remainder theory, we'll get f 2 is equal to 4 So a 2 cube plus b 2 square plus 2 minus 6 which we must be equal to 4 So 8 a plus 4 b minus 4 is equal to 4 after simplification and dividing the entire equation by 4 you'll get this much Simplifying you can say this is 2 a Plus b is equal to 2 This is equation number 2 Or you can write 2 a plus b minus 2 is equal to 0. This is equation number 2 Now let's rewrite both the equations together. So we get 2 a minus b plus 2 is equal to 0 and We get 2 a plus b minus 2 is equal to 0 guys Okay, so you'll get this is 2 now if you add both these equations that is I am doing 1 Plus 2 that means LHS of 1 plus RHS of 1 is equal to sorry LHS of 1 plus LHS of 2 Will be equal to RHS of 1 plus RHS of 2. So let's add them together What will you get 2 a plus 2 a will get you 4 a minus b plus b is 0 Plus 2 minus 2 is again 0 and this entire thing is 0. That means a has to be 0 so we get a 0 So if a 0 we can deploy in any of these equations. So in the first equation if you see from 1 Minus b plus 2 will be 0 because a was 0. So hence b is equal to 2 Okay, so we could find out the value of a and Value of 2 what did we apply or what was the underlying concept first? We applied factor theorem found out one equation then applied remainder theorem found out the second equation solve the two equations in linear equations and We got the value of a and b. So while solving this linear equation. Please remember how to solve this is you know Adding and substitution adding and subtracting may help there is another way of Solving linear equations in two variables called the substitution method Which you can have a look on our course on linear equations in two variables