 Hello and welcome to another session on polynomials and in today's session we are going to discuss another very important theorem utilized in problem solving related to polynomials and in many other aspects of mathematics and that is factor theorem. Now in the last session we saw some examples and problems on remainder theorem. Factor theorem can be considered to be the extension of remainder theorem. So we'll first recap the remainder theorem and we'll see how one special case of remainder theorem gets converted into factor theorem. So we had learned remainder theorem in the previous sessions and we saw some examples also and what was remainder theorem. It said that when a polynomial is divided by another linear polynomial ax plus b. Then the remainder in such division, the remainder you will get is nothing but let's say this polynomial was f of x. When a polynomial fx is divided by ax plus b, then the remainder is f of minus b by a. That is what was remainder theorem and hence let's say the expression or the equation becomes fx is equal to ax plus b times gx plus the remainder rx. Now everything is a polynomial in x but yes in this case remainder is a constant term but just for simplicity and uniformity sake we will be writing rx as well because any polynomial, a constant polynomial can also be considered as a polynomial in x where the degree is zero. So fx is ax plus b gx plus rx. Now this is how we used to find out remainder. So how did we find out remainder? Remander was simply minus b by f of minus b by a. F of minus b by a will make this as a into minus b by a plus b. So wherever there is x you have to simply replace it by g of minus b by a. So for some polynomial g plus remainder r I am just simply writing r because that's what I am interested in finding out. This is what was from remainder theorem. Now special case. We are going to talk about a special case. What is that special case guys? So what happens if r becomes zero? Let's say this becomes zero. Okay, so remainder is not zero. So remainder is zero sorry. So if remainder is zero if remainder is zero in any division process then the divisor is a factor of, becomes a factor is it? If remainder is zero then divisor is a factor. Example if 20 is equal to 5 into 4 plus 0 that means remainder if you divide 20 by 5 you will get quotient as 4 and remainder as 0. This was q and this is r. Then clearly 5 is a factor of 20 and we write 5 divides 20 like that. Isn't it? 5 is a factor of 20. Okay, so this sign is 5 divides 20 or 5 is, this is the symbol or the notation 5 divides 20. 5 is a factor of 20. This means 5 is a factor of 20. Similarly if this remainder here becomes zero this is becoming zero then what will be the, you know, the expression or equation is fx is equal to ax plus b gx. Reminder is zero so this will be like that. So if r is zero and what was r by remainder theorem? By remainder theorem r remainder for any such division is f of minus b by a when the divisor is ax plus b. Isn't it? So if r is zero that is if minus b by a is equal to zero then ax plus b is a factor of fx. This is what is factor theorem. Okay, special case of remainder theorem understood or vice versa. If the other way round is also true if ax plus b is a factor of ax plus b is a factor of fx then f of minus b by a will simply be zero. Right? One on the same thing why f of minus b by a by remainder theorem is a remainder obtained when the fx is divided by ax plus b. Now if ax plus b is a factor then obviously remainder will be zero so hence this is the condition that if the polynomial ax plus b is a factor of fx then f of minus b by a is equal to zero. Let's take an example and understand. Let's say fx is 2x square minus 7x plus 6. Okay, let's say fx is this. Now we have to check if x minus 3 is a factor of fx. So if fx minus 3 is a factor of fx then f of 3 must be equal to zero. Right? Here if you compare with our ax plus b so ax plus b this is the linear divisor. Here my ax plus b is x minus 3 so a is equal to 1 and b is equal to minus 3. Correct? So minus b by a is 3. Okay? So hence remainder will be simply f of minus b by a that is f of 3. Isn't it? So f of 3 is the remainder. Right? So f of 3 let's check f of 3 is 2 into 3 square minus 7 into 3 plus 6. What is it? This is 18 minus 21. So 18 minus 21 plus 6. How much is it? Check. If you check this is 3 which is not equal to zero. Not equal to zero. Therefore, therefore we can say that x minus 3 is not a, so let me put a question mark here because this is not certain. So x minus 3 is not a factor of fx. Okay? So this is we conclude without actually dividing it with x minus 3 and checking whether we are getting remainder zero or not. You can directly say that x minus 3 is not a factor of fx. In this example we take again fx is equal to 2x square minus 7x plus 6. Okay? Now let's check if x minus 2 is a factor of fx. Okay? So let's check. So we have to again put f equals f2 to check. Right? Again, if you do the calculations here, ax plus b you have got x minus 2. So a is 1, b is minus 2. So minus b by a has to be 2. So you have to now check for f of minus b by a that is f of 2. So let's check f of 2. So f of 2 is equal to 2 into 2 square minus 7 into 2 plus 6 which is 8 minus 14 plus 6, 0. Therefore, since f2, so you write since f of 2 is 0, therefore, x minus 2 is a factor of fx. Correct? Right? And fx was 2x square minus 7x plus 6. Right? The moment you see that f of any number is becoming 0, that means x minus that number will be factor of fx. Correct? So this is what is factor theorem. Let's try and solve some problems related to factor theorem.