hi guys hi guys hi guys hi guys hi guys it's me teacher in today's video hi guys it's me teacher going in today's video we will talk about how to find the zeros of a polynomial function when we say zeros of polynomial function we are trying to calculate or find an input or values of x that can turn the polynomial function b equal to zero so without further ado let's do this topic so we have here zeros of polynomial function and a main target attendee is to use rational zero theorem we are given p of x is equal to x cubed plus 2x squared minus 5x minus 6 where in this one is a cubic polynomial function now target method is to solve or to find the zeros of this function using the student so when you say rational zero theorem value number e over q where in pinaten it represents the different factors of your constant term factors of your constant term and the factors of your leading coefficient oxidability leading coefficient this this is the coefficient of your uh leading term wherein the term has the highest exponent changing the dilemma degree pneumatic function now um this is starting to step by step and first step is to prepare all the possible factors of your constant verb so before i include a new value over q for the factors of your constant term expingula we have here negative six okay the factors of negative six are one negative one two negative two three negative three and six comma negative six and to simplify this guys and that factors have been added uh these are numbers that when you multiply them it will give you this product so exactly one times negative six is equal to negative six negative one times six is negative six two times negative three is negative six and negative two times three is negative six but to simplify without writing starting different [Music] as positive negative 1 comma positive negative 2 comma positive negative 3 comma positive negative 6 and as you can see we are done with the possible factors of your constant verb and now let's move on with the factors of your q or the leading coefficient which is one since one thumbnail of possible factors positive or negative one one times one is equal to one and negative one times negative one is positive one so they tell me the domain guys i we will simplify this star q in fact first this will give us all the possible values of x or sorry all the possible zeros of this function remember you're given at the speed of x is equal to 1 is equal to x cubed plus 2x squared minus 5x minus 6. since cubic you're adding polynomial function um we're expecting three possible zeros melon time that long possible zeros okay once again here we have p over q oh simplify that into guys uh this will give us um p over q is equal to positive negative one over positive negative one where in by simplifying the n this will give you the answer of positive negative one and when you simplify it positive negative two by positive negative one in the beginning you try positive since you add in denominator i positive negative y and the total name again positive negative three positive negative six so all in all you you my possibility is nothing to get the zeros at all one two three four five six seven eight melon time six options in that part what is possible zeros because we are in the cubic polynomial function so what is the first step ah basically i will start with the positive one aka starts a positive one in your textbooks and some google search but in my in our case i'll be using the remainder theorem if i'm gonna use a value of x here as my input from this part eight thousand eight possibilities you will remain international is equal to zero automatic one of them is a zero of the function so i will try to use f of what okay i will use f of one negative you negative one if candidate zero again so i will use this f y in which i will copy the given function we have x cubed plus two x squared minus five x minus six again i am used using remainders together we're in if input put on one starting given function and this will turn out as zero automatic one is one of the possible zeros of the function so let's try this is 1 cubed plus 2 times 1 squared again i'm replacing the x with the value of 1 minus 5 times 1 minus six and simplifying this powers or exponents you can get one out of one cubed plus one squared is two times two that is two again one squared is one times two is two and this one we have negative 5 times 1 this will give us negative 5 minus six okay and simplifying this this will give us f of one is equal to this is negative as you can see the remainder here is negative eight it becomes your one nut and it's not zero and this is zero in one okay so let's try another i will after using positive one i'll be using negative one so i got that f of negative one is equal to i will directly replace the lambda power of zero space f of negative of one is equal to negative eight so we will stop here and i'll be using f of negative one f of negative one is equal to negative one cubed it replace another plus 2 times negative 1 squared minus 5 times negative 1 plus six now that one negative one could not then since the exponent is odd this is equivalent to negative one and for this part negative one squared is positive one times two this will give us the answer of plus two and paragraph ditto negative five times negative one this will give us the answer positive five and then plus six okay so when you simplify this your f of negative 1 is equal to eleven this is twelve so independent zero so as you can see guys in deeper in zero you're not going to remain there the possible um positive ones are negative y so next time we have an option economic penalty possibility we will be using positive two so we will use f of 2 replace that again your x here will be positive 2 so replace that and we have 2 base 3 plus two times two squared minus five x minus six so happens in here at the replacement x naught and b number two so this is negative five times two minus 6 and adding 2 cube this one is equivalent to 8 your 2 raised to 2 is equal to 4 times 2 this will give you plus 8. and negative five times two this will give us negative ten we have negative six so simplifying this this will give us eight times eight plus eighty sixteen minus ten that is six minus six seven f is one of the zeros of the given function we have x is equal to 2. um we have three possibilities or three possible zeros of this function we are done with the first zeros with the first zero sun target is to find the other two pero um is to use the synthetic deviation okay synthetic deviation we have x is equal to two we will use the coefficients of this one negative two we have negative five we have negative six where in by using synthetic deviation drop down at the top we have one times two that is two and then two plus two is 4 4 times 2 that is negative that is 8 so we will subtract it we have negative 5 plus 8 this is 3 3 times two that is six and negative six plus six this will give you the answer of zero if it's obvious this one is x squared plus four x plus three at the remainder not gonna be zero uh this is a zero of a function now this will become x squared plus four x plus three is equal to zero we're in we can use the quadratic way on how to solve this regarding factoring we're going to factor out x squared plus four x plus three the factors of three to make it four that is x plus three times x plus one is equal to zero so three times one is three three plus one is four so we can use the zero product property to equate each factor by zero we have x plus three is equal to zero and the other is x plus one is equal to zero theta transpose will give us x is equal to negative three and then as possible we have x is equal to negative y and as you can see those of the functions is equal to two as the first zero we have x is equal to negative three we have x is equal to negative one and the solution here x and zero n are this negative one and two this solution will represent the possible zeros of this polynomial function so i hope gay standard authorize the first partner on how to solve or how to algorith for the zeros of the polynomial function in the next part of our video when we begin processing another example we're in same process now here's our example number two we are given the function f of x is equal to x to the fourth power minus five x cubed plus 20x minus 16. now this one is a polynomial function with the degree of four and let's try finding the value of p over q again your p are the factors of your constant which is positive negative 1 positive negative 2 positive negative 4 positive negative v tap was positive negative 16 and as for the value of q a new fat person leading coefficient which is one so we have positive negative y and simplify the possible values of p over q that is positive negative one positive negative two 2 positive negative 4 positive negative 8 positive negative 16 so possibilities we have 1 2 3 4 5 6 7 8 9 10 we have 10 possible zeros of x and since you're in degree a4 we are expecting four possible zeros of this function so subject we will use synthetic division so positive one so i will use positive one and using synthetic division i'll be getting the coefficients of this function the first is one and this is negative five and when i notice quadratic term it makes at the end that is zero and then we have twenty tapos minus 16. so we will start the use of the synthetic division okay we have to bring down one one times one is one negative 5 times 1 up plus 1 is negative 4 then negative 4 times 1 that is negative 4 0 plus negative 4 is still negative four okay i'm at negative four then we have here 20 uh negative four times one that is negative 4 we have 20 plus negative 4 that is 16 and we have here 16 times 1 that is positive 16 and adding these two numbers or integers this will give us zero as you can see this is our remainder zero it disappears guys one of the zeros one of the zeros is we have here x to the third power since porto minus final one i'm again leading terminator then minus four x squared minus 4x plus 16 n and again in first zero nothing is equal to 1. so given this function we can convert this into an equation we have zero or going to minimum x minus four x squared minus four x plus sixteen is equal to zero equations we will try to factor out x cubed minus 4x squared minus 4x plus 16 parametrium three other zeros of this function i would recommend now factoring by grouping so we will try so this is x minus one factor then we have x cubed minus four x squared okay and then plus negative four x plus sixteen in a group and then we can factor out something here in fact which is x squared so we have x minus one then this is x squared times x minus four so x squared times x minus four that and it came from x cubed minus four x squared and finally though we have um we can factor out negative four so we have negative four times x minus four so neon original equation so as you can see has so we can still factor out this one x minus one for leading times x squared minus four and examinating x squared minus four and then since common though we have another factor which is x minus four is equal to zero we haven't we can still factor out x squared minus four let's say a difference of two squares the factors of this are the next one x minus one times x plus two x minus two getting factors in little x squared minus four and n times x minus four is equal to zero as you can see we have four different factors at something now we have four possible zeros of this function and right now we will solve for the values of x una first factor we have x minus one is equal to zero equate that into that zero place we have x is equal to y this is the first zero transpose this to the other side from negative positive next we have x plus 2 is equal to zero transpose this to the other side will give you x is equal to negative two this is the second zero of the function next then we have x minus two is equal to zero transposing this to the other side this will give us x is equal to positive two at an end third possible zero lastly we have x minus four is equal to 0 transpose this to the other side will give us x is equal to 4. you possible zeros of this function and write down our solution or zeros we can say that we have negative two comma one comma two comma four as zeros of the given we will use synthetic division so if you're new to my channel don't forget to like and subscribe but hit that bell button for you to be updated exactly latest uploads again it's miniature going on bye