Good day students welcome to mathgotserved.com in this clip where going to be going over the factor and remainder theorem's before we get started with the problem-solving process we are going to take a look at what's the factor in remainder your arms are little starts with the remainder theorem there remainder there know what does the remainder theorem say oh this is what it says is a polynomial if the polynomial let's see the polynomial is FO backs in that polynomial F of X is divided divided by the binomial X minus R then the remainder is this colder remainder of the case are okay the remainder is F of our okay so what is this theorem telling us if we're dividing the polynomial by a binomial of this form where a is equal to one on we can simply find the remainder by fine in the zero of that divisor in this substance used in it into all the exes in your dividend polynomial function okay so in essence you do not really have to divide out the polynomial either by synthetic or long division in order to determing what the remainder is okay so that's basically the remainder there now all the remainder the romantic the remainder theorem have a special case the special case of the remainder theorem is the factor theorem now what happens when you remainder is zero that gives us the factor theorem so the factor theorem is as follows a polynomial let's say FO backs have a factor have a factor of the form X minus R is and only if the function and evaluated at the zero of that oh factor F of our is equal to zero okay with this is same in essence is that the remainder F of our athlete indicated above is equal to zero this is what a factor is even though mainly factor of another number when you divide that number by that's factor you do not have any remainders okay now let's take a look at an example okay example number one we are to find the remainder if the polynomial function X to the third minus 7X square +20 X -22 is divided by X -4 okay so find the remainder is this polynomial function is dividend is divided by this divisor in three different ways okay in this lesson will going to be reviewing two other ways to find the remainder namely the synthetic division process and the long division process okay so in three different ways that the first parts and then you have to answer the question is this divisor right here X minus one is X -4 a factor of the dividend is it okay right let's start with all parts one which is to find the remainder okay to find the remainder method one that were going to use is the factor know the remainder theorem okay the method one is remainder theorem so in on its user remainder theorem first of all we have to identify what's be Om divisor is okay so take a divisor which is X -4 we said it equal to zero in little find the zero this divisor okay so back to be accomplished by simply adding four to both sides of the equation so X equals four is a root of this equation okay now what were going to do next were simply going to evaluate this divisor polynomial the dividend polynomial sorry by the roots of the divisor which is four okay so FO backs is X to the third minus 7X square +20 X -22 so if you remember the remainder theorem would you simply about we dysfunction at the zero of the of divisor so we're going to go F of the county X replace with four FO for is equal to apprentices for race to the third power -7 times for racer the second power +20 times for -22 okay so what will do now is simply plug this entire expression in zero calculators if you do so you end up with the value of 10 okay while the of 10 let's up the registry with the calculator a quick so what we're looking at here is four to the third power -7 times for recent the second power +20 times for -22 and we get 10 okay so what if we just find we have just found the remainder rights of the remainder is equal to 10 when you divide this function right here by X -4 so it easy how factor was to compute the remainder using the remainder theorem especially if you have a calculator in your possession now let's take a look at method to which these three methods method to is synthetic division okay so let's do that method to is synthetic division okay so for synthetic division we have to just set up the polynomial correctly first and then now we can go on from there now let's set up our synthetic division bars within a place the coefficients of the function of the the coefficients of the the the dividend function in the division bars is such a way that no missing degree term her her our included okay so we don't I have any missing degree terms so we have 3210 in this case no degree term is Mrs. of that's perfect had it been we had a mission term will have to use a placeholder method applicable here okay so there's a one in front of the function so we have one -720 -22 and then we are going to place the zero of the divisor polynomial which is for right there in there
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