good day students welcome to mathgotserved.comin this clip where going to be going over the factor and remainder theorem's beforewe get started with the problem-solving process we are going to take a look at what's thefactor in remainder your arms are little starts with the remainder theorem there remainderthere know what does the remainder theorem say oh this is what it says is a polynomialif the polynomial let's see the polynomial is fo backs in that polynomial f of x is divideddivided by the binomial x minus r then the remainder is this colder remainder of thecase are okay the remainder is f of our okay so what is this theorem telling us if we'redividing the polynomial by a binomial of this form where a is equal to one on we can simplyfind the remainder by fine in the zero of
that divisor in this substance used in itinto all the exes in your dividend polynomial function okay so in essence you do not reallyhave to divide out the polynomial either by synthetic or long division in order to determingwhat the remainder is okay so that's basically the remainder there now all the remainderthe romantic the remainder theorem have a special case the special case of the remaindertheorem is the factor theorem now what happens when you remainder is zero that gives us thefactor 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 functionand evaluated at the zero of that oh factor f of our is equal to zero okay with this issame in essence is that the remainder f of
our athlete indicated above is equal to zerothis is what a factor is even though mainly factor of another number when you divide thatnumber by that's factor you do not have any remainders okay now let's take a look at anexample okay example number one we are to find the remainder if the polynomial functionx to the third minus 7x square +20 x -22 is divided by x -4 okay so find the remainderis this polynomial function is dividend is divided by this divisor in three differentways okay in this lesson will going to be reviewing two other ways to find the remaindernamely the synthetic division process and the long division process okay so in threedifferent ways that the first parts and then you have to answer the question is this divisorright here x minus one is x -4 a factor of
the dividend is it okay right let's startwith all parts one which is to find the remainder okay to find the remainder method one thatwere going to use is the factor know the remainder theorem okay the method one is remainder theoremso in on its user remainder theorem first of all we have to identify what's be om divisoris okay so take a divisor which is x -4 we said it equal to zero in little find the zerothis divisor okay so back to be accomplished by simply adding four to both sides of theequation so x equals four is a root of this equation okay now what were going to do nextwere simply going to evaluate this divisor polynomial the dividend polynomial sorry bythe 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 dysfunctionat the zero of the of divisor so we're going to go f of the county x replace with fourfo for is equal to apprentices for race to the third power -7 times for racer the secondpower +20 times for -22 okay so what will do now is simply plug this entire expressionin zero calculators if you do so you end up with the value of 10 okay while the of 10let's up the registry with the calculator a quick so what we're looking at here is fourto the third power -7 times for recent the second power +20 times for -22 and we get10 okay so what if we just find we have just found the remainder rights of the remainderis equal to 10 when you divide this function right here by x -4 so it easy how factor wasto compute the remainder using the remainder
theorem especially if you have a calculatorin your possession now let's take a look at method to which these three methods methodto 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 firstand then now we can go on from there now let's set up our synthetic division bars withina place the coefficients of the function of the the coefficients of the the the dividendfunction in the division bars is such a way that no missing degree term her her our includedokay so we don't i have any missing degree terms so we have 3210 in this case no degreeterm is mrs. of that's perfect had it been we had a mission term will have to use a placeholdermethod 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 rightthere in there we're going to carryout our synthetic division not recall from previouslessons you divide synthetically by adding downwards the multiply and across there isno number here so start with zero okay so i down want to zero is one in there we'llto multiply across four times one is for at down again you -3 multiply for by -3 you -12at down the get eight multiply 4ã—8 you get 32 i down any end up with positive 10 of hislast digits underneath your division bars is your remainder as you can see our remainderhere equal to 10 and that agrees with our results in method number one on the last methodwill going to use to find the remainder is
the long division method okay so let's goahead and do that method three long division okay long division this is the longest methodthink about the name on so let's go ahead and set it up we have our long division barswill extract the dividend function exactly as it is x to the third minus 7x square +20x -22 now you also want to follow the same guidelines out the synthetic division processgive any degree terms are missing you have two put a placeholder there okay so we have3210f indicated earlier there the missing degree terms when you descending from thehighest degree to the constant so this is fine now we divided by om x -4 so let's playso x -4 out here okay so how do we do long division what will do is eat we just focuson the medium-term of the divisor in the medium-term
of whatever sections of the dividend polynomialwe are factoring okay so when i'll looking at the medium-term of this original dividendpolynomial and deleting term of our divisor okay and ask yourself how many times dollarsx going x to the third x goes into x the third x square times because x square times x isx to the third okay so multiply that out to multiply we have x to the third x were templatedfor is negative for x square okay now what will do is will proceed to subtract rememberdivision is use repeated subtraction you repeatedly extract of the factor from the the dividendin a number of times you can do that is basically a quotient okay so let's extract this factorput a minus when he carryout to subtraction the first terms always counsel out that'sby design okay so x to the third minus x to
the third able to zero negative x7x squareminus minus minus minus laplace so you 7x were plus 4x square is negative 3x squarebe seen as -3 right here that's the same as is -3 in the synthetic division algorithmokay now all will going to do next is bring down another terms since we our divisor isa binomial are going to be bringing them down one at a time so will have two terms and everytime if we eliminate this for example would bring down to okay so we have +20 x what weregoing to do now is repeats the same procedure how many times is x go into the leading termhere are incomes is x going to negative 3x square or x times what gives you negative3x square the answer is negative 3x okay they are 3x times x is unita 3x square of a changethe color there is a psalm negative 3x square
and then on -3 times for gives us positive12 x and alan going to om subtract this factor again when he carryout those subtraction whatdo we get if we carry out the subtraction we have all these two counsel out the addup to zero swaney positive 20 minus +12 minus terms plus is minus to have 20-12 which is8x and then will bring down the last terms is redoing with to out a time okay so bringdown -22 okay now we're going to carryout this procedure one last time how many timesdaws x going to 8x and goes into 8x positive eight times okay so and out down to the eighthright here in our synthetic division that's bz the where we are again now multiply thattimes x is 8x8 times negative for is -32 and we subtract that factor again remember divisionis repeated subtraction extraction the factors
so if we carryout the difference that willgive us to remainder that's we are looking for so eight minus 88x minus 8x is zero theof 22 minus -32 - minus laplace so have -22+32 that gives us 10 and that ladies and gentlemenis our remainder okay so if you take a look at these three methods that we use just nowwhich is the most efficient methods for finding the remainder if you have a calculator orand your good with your order of operations remainder here is lonely the fastest okayand the synthetic division is not too far behind sometimes this might actually be betterso what's the significance what the relevance of methods two and three will methods twoand three do not just tell you the remainder the also tell you the quotient okay syntheticdivision told the quotient if you take a look
at these other coefficients of the depressedpolynomial of the quotient which is let's write it down omelettes see one x square isa degree list of the original what x square minus 3x +8 and then we have the remainder10 over x -4 okay so these two tell you not just the remainder box the also toledo quotientnow to think a look of these two methods synthetic a long division which one is the better methodfor dividing polynomials well it depends synthetic division is known for his speed is very veryfast and long division is powerful okay now widely vein why do we ascribe these qualitiesto be so methods will synthetic division if you are presented look probable certain secondlet's see the the divisor is of the form x minus our then it's really easy to use syntheticdivision to dividing gets her answer but if
you have a divisor that's of the form let'ssay a x square plus be x plus c are if you have a x minus our if the divisor is not ofthis form then this method could fall apart or you can be very very difficult to use okayso it is presented in this form you definitely want to use synthetic division but in allthe forms you end up having to rely on the long division so that's why this method ispowerful because it can handle any kind of division of polynomial problem you throw itsway but this one is limited when it you present problems that of a particular second thatit can handle it's extremely fast to use okay so you it's important that you know all threemethods in you also want to know the best methods to use for any particular problemthanks so much for taking the time to watch
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