 Hello and welcome to another problem-solving session on factor theorem the question says Find the value of k if x minus k is a factor of fx and Fx is given as x cube minus k square x plus x plus 2. Okay So, you know, so they are you know any way suggesting that factor of fx that means we have to use factor theorem And what is factor theorem guys? factor theorem from factor theorem we can say What can we say that if ax plus b is a factor of a polynomial fx then F of minus b by a is equal to 0 Correct. This is what we have learned now here. Let's compare ax plus b so ax plus b is a factor of is a factor of Right, this affects. What is fx? x cube minus k square x plus x plus 2, okay, and x ax plus b in our case is this in our case is x minus k So clearly a is 1 b is minus k So friends minus b upon a is simply k Isn't it? So let's check fk. So if x minus k is a factor of Fx which affects the given effects here then Then f of k must be equal to 0 So let's find out. So what is f of k f of k will be simply k cube just replace x by k So k square times k Plus k plus 2 must be 0 Now clearly k cube and k square k is k cube so gone. So this implies K is equal to minus 2 So if k is equal to minus 2 then x cube minus Or x cube plus 2 x square or right. So you replace k by minus 2 basically. So whatever is the value So k has to be minus 2 for this particular expression to be divisible completely by ax plus b or For ax plus b to be a factor of this particular polynomial k must be equal to minus 2 Now let's look at this question says find the values of a and b so that the polynomial fx is equal to x cube plus 10 x square plus ax plus b is Exactly divisible by x minus 1 as well as x minus 2. So again, we'll use factor theorem. So you can say by factor theorem by factor theorem what will happen if x minus 1 is a factor of Fx Right, then f of 1 is equal to 0. That's what we have learned right So f of 1 it should be 0. So let's put f of 1 to be 0 You know, I have a you know habit of writing the general factor theorem. So ax plus b divides or is a factor of fx any polynomial fx if f of minus b by a is equal to 0 This is factor that I mean not sure Right. So in this case, you will say, okay, where is a and b in this case? So the linear expression is linear divisor is x minus 1 in our case c x minus 1 So if you compare this x minus 1 with our ax plus b Which is our general from theorem general thing from theorem then what is a a is 1 isn't it coefficient of x What is b? b is minus 1 by the constant term. So minus b by a simply 1 So hence you see directly I have written f of 1 is equal to 0 f of minus b by a must be 0 So minus b by a in this case Is 1 so hence f of 1 must be 0 So I mean again, I am talking only in this term because this is the this one is the general form So you must remember this one rather than any specific form So for that matter tomorrow, there is another expression 3x minus 5 or 7x minus 2 Whatever, then you know what to do in that case, right? So what will be how to find out whether something is a factor of Fx or not so in this case clearly we have to simply check f 1 is 0 or not So but then it is given that x minus 1 is a factor. So definitely f of 1 must be 0 So hence what is f of 1 deploy 1 in the FX that is you have to find out f 1 and Value of FX at x equals to 1 will be simply 1 cube. So replace x by 1 10 times 1 squared Plus a times 1 plus b is 0. So you get a Yeah, so hence if you simplify 1 Plus 10 11. So a plus b plus 11 is 0. This is equation number 1 write it like this equation number 1 Next x minus 2 is also a factor. So you write since x minus 2 is a factor of FX Then again, what do we know F of 2 will also be 0 Right, so let's write f of 2. So f of 2 is 2 to the power 3 Plus 10 to square 10 times 2 square plus 2 a plus b and this is also 0 So let's solve it. This is 8 plus 40 10 into 2 square is 40 plus 2 a Plus b is 0. So that means 2 a Plus b plus 48 is 0 equation number 2 what was equation number 1? Let's write that here a Plus b plus 11 is equal to 0 Okay, now this is called a pair of linear equation in two variables a and b are two variables So hence many of you would have not Encountered this before you might worry that how to solve this. It is very simple You know, you can deploy this trick LHS minus LHS Is equal to RHS minus RHS. So subtract these two equation. That means I'm doing two Minus one. I'm subtracting second first equation from the second equation So you have to simply subtract LHS from LHS and RHS from RHS. So let us subtract LHS from LHS first So 2 a minus a this will become minus. This will become minus So 2 a minus a is a b and b will go 48 minus 11 is plus 37 and RHS minus RHS is 0 Is it so you understood how to solve these two equations? LHS minus LHS and RHS minus RHS you did and you get a Plus 37 is equal to 0. So a will be clearly minus 37 Right, so you got a Now for B, you can use any of the equations to find B, right? So let's say we are taking one So a plus B plus 11 is 0. So a we just found out minus 37 Plus B plus 11 is 0 So what will be B? So B is equal to 37 minus 11 is equal to 26 Is it it? So hence what do we get? We get a is equal to minus 37 and B is equal to 26 Now these are the values for the given condition to be fulfilled, right? So this is how factor theorem can be applied to find out Different, you know, uh missing coefficients in a polynomial