hi friends are you ready to master the topic of polynomials then this video is for you because in this video we are going to be looking at the key concepts of polynomials and we'll be solving some important questions on this topic I'm going to show you the tips and tricks on how to solve the sums and if you find this video useful to hit the like button and share it out with your friends and do check out my website Manoj academy.com for more questions for you to practice and friends as you know practice makes you perfect so I want you to pause the video right here and go and get a pen and paper so that we can solve the polynomials topic together so let's dive right into it let's start with the first important concept which is if px is a polynomial zeros of the polynomial are those values of X okay which make the polynomial 0 for example let's take this polynomial px equal to x squared minus 5x plus 6 now to find the zeros of this polynomial we need to set this polynomial as 0 so this will be our equation x squared minus 5x plus 6 equal to 0 now to solve this equation we can factorize the left hand side so let's do middle term breaking so this is going to be x squared minus 3x minus 2x plus 6 equal to 0 right and now can you see the factors so if you factorize this you are going to get X minus 3 multiplied by X minus 2 equals 0 so either X minus 3 is 0 or X minus 2 is 0 right which means the values of X are going to be 2 or 3 right so the use of X are going to make the polynomial zero right and these values of X we said they are called the zeros so these values of X are called the zeros or they're also known as the roots of the polynomial so you need to know these two terms they're either called the zeros or the roots so it's very simple so to find the zeros we need to make the polynomial equal to 0 so polynomial px is set equal to 0 that's our equation and the solution of that gives us the zeros or the roots which is those values of x that makes the polynomial 0 now let's look at the next concept the relation between zeros and coefficients of a quadratic polynomial so let's say a quadratic polynomial px equals ax squared plus BX plus C where the coefficients are a B and C right and the coefficient a is not equal to 0 why because we want a quadratic polynomial right so we want that term x square in there so a should not be 0 okay and let's say that alpha and beta are the zeros or the roots of the quadratic polynomial px then there's this important relation between the roots and the coefficients so between the zeros and the coefficients and what is it that alpha plus beta is equal to minus B by a so the sum of the roots is minus B by a these coefficients and the product of the zeros is alpha beta is equal to C by a so these coefficients okay so note these two important relations between the zeros and the coefficients of the quadratic polynomial and if we know the zeros of the polynomial alpha beta then we can write the quadratic polynomial as X square - alpha plus beta into X plus alpha beta right so basically it's X square minus some of the roots times X plus the product of the roots so we can write the quadratic polynomial in terms of the zeros here so note this important formula now let's take a look at this question obtain the zeros of the polynomial root 3x squared minus 8x plus 4 root 3 and verify the relation between its zeros and coefficients okay so remember we are going to copy down that polynomial and to find the zeros we need to set it to 0 so it's root 3x square right minus 8x plus 4 root 3 and we'll set it equal to 0 okay and now we can use middle term braking to factorize this polynomial so can you see that the sum is going to be minus 8 and the product is going to be 4 root 3 into root 3 so that's 12 so let's break the middle term like this root 3x squared minus 6x minus 2x plus 4 root 3 equals 0 so can you see that the sum is minus 8 and the product 6 into 2 is 12 okay so now let's factorize this so it's going to be root 3x and this is going to be X minus two root three and here we'll have a minus two so X minus two root three again equal to zero so if we group this we are going to get the factors as root 3 X minus 2 into X minus 2 root 3 equals 0 ok so what are the roots here or what are the zeros so X is going to be so either this is 0 or this is 0 and so we have x equals 2 by root 3 from this one right and from this one also we get X equal to 2 root 3 right so these are the zeros of the polynomial of our quadratic polynomial right so we found the zeros here this is our answer and now we need to verify the relation between the zeros and the coefficients so remember the formula that we learnt alpha plus beta equals minus B by an alpha plus alpha beta equals C by so let's apply that so what is our formula here so let's write that down alpha plus beta equals minus B by a right the sum of the roots of the sum of the zeros and so that's going to be minus B by a so what is the coefficient so if we are going to compare this with remember what is the standard form with the polynomial quadratic polynomial ax squared plus BX plus C right and that we had set equal to 0 so here B is 8 sorry minus 8 and is root 3 right so this is going to be minus off we have B as minus 8 here and is root 3 so what is the sum of the zeros going to be it's going to be eight very root three okay and now let's do the product of the roots alpha beta the product of the zeros or the roots is going to be CY that's our formula right so again we need to use our coefficient cn a so what is the coefficient C it's basically four root three right and is root 3 root 3 cancels and the product of the zeros is four okay and so to verify this relation because we have used this relation right alpha plus beta and alpha beta since we know the zeros so we are going to use it and check if these values that we got using the relation are they right or not okay so that's pretty simple so first thing you'll do is take alpha plus beta let's do that so that's going to be 2 by root 3 plus 2 root 3 right ok and if we take the LCM as root 3 okay so 2 plus 2 root 3 into root 3 that's going to be 6 and we are gonna get ate by root 3 okay so do we have the correct answer yeah there you can see alpha plus beta is 8 marry root 3 and that's what we had got from our relation minus B by a so we verified that and now let's quickly verify the product of the zeros right so we are going to solve for alpha beta and that's pretty simple because it's going to be 2 root 3 into oh sorry 2 by root 3 into 2 root 3 okay and so the root 3 gets cancelled there and we get 4 so here if you take a look we got alpha beta by using the zeroes as 4 and that matches exactly what we got in our relation so that's the main thing here in this question first you we use factorization and we solved for the zeros and then we used our formula alpha plus beta minus B by alpha beta C by and got the values and verified that from the roots itself and so you can see that it's matching so the relation between the zeros and the coefficients is also verified here so it's pretty simple right are you ready for the next question here it is if X equal to 2 by 3 and X equal to minus 3 are the roots of ax squared plus 7x plus B equal to 0 we need to find a and B so how do we solve this remember the concept if these are the roots or the zeros that means these values if we substitute it in our polynomial ax squared plus 7x plus B is going to be 0 which means it's going to satisfy this equation here so let's write down our equation which is ax squared plus 7x right plus B equal to 0 and now let's simply substitute these values so let's substitute the first value is going to be a into 2 by 3 whole square right plus 7 into 2 by 3 plus B equals 0 so if we simplify that we will get 4 by 9 8 plus 14 by 3 B plus 14 by 3 sorry plus B equals 0 ok so this is our first equation and now let's substitute the next value X equal to minus 3 again in this equation right so what do we get we are going to get a into minus 3 whole square plus 7 into minus 3 plus B equals 0 right and so if we simplify this we get 9a minus 21 right plus B equal to 0 and that's our second equation here ok and we the rearrange and write this equation as 4x9 a plus B equals -14 by three right so that was our first equation rearranged and similarly here we have nine a plus B equals 21 okay so now it's pretty simple because we need to find a and B and as you can see we have two equations so right so we have a pair of linear equations two linear equations in two variables so you can simply solve that and if you go ahead and solve it you're going to get the values so try this out yourself you're going to get a equal to three and B equal to minus six so that's going to be our answer if you solve these two equations note this important point that if a is a zero of a polynomial P X then X minus a divides P X or we can say X minus a is a factor of P X for example let's say 2 is a zero for polynomial P X then we can say that X minus 2 divides P X or X minus 2 is a factor of P X now let's solve this question find all the zeros of 2x cubed plus x squared minus 6x minus 3 if two of its zeros are root 3 and minus root 3 so how do we do this question first let's write down our polynomial so can you see it's a cubic polynomial here so let's write that down so let's say the polynomial FX is 2 X cube plus X square minus 6 X minus 3 okay and since we want to find the zeros of the polynomial we are going to equate the polynomial as 0 right now we've been given that we have these two zeros root 3 and minus root 3 okay so we can say that X minus the zeroes of the roots so it's going to be X minus root 3 and X minus of minus root 3 will be X plus root 3 so can we say that these are factors of FX right so these are factors of FX because they divide it right because these are the zeros of this polynomial so now if we multiply this so it's going to be you can use a plus B into a minus B and it's going to be X square minus 3 multiplying these two out we get X square minus 3 so X square minus 3 is a factor of this polynomial FX that means this is this polynomial can be divided by this right so if we divide these two so if we divide the polynomial by X square minus 3 so let's write it like this so X square minus 3 if you sit and do this division so 2x cubed plus x squared minus 6x minus 3 right so if you sit and divide this and you solve this you're going to get the quotient as 2x plus 1 okay so now how can we find the other zero so we've got the quotient as 2x plus 1 this means we can write this polynomial FX as FX equals x square minus 3 multiplied Bar quotient here which is 2x plus 1 right so we're going to multiply this right and this is equal to 0 I'm just going to draw a line here right to separate this and this is equal to 0 right which means that this factor 2x plus 1 is equal to zero so we can say if we have two x plus one equal to zero we can write that down here 2x plus 1 is 0 therefore X equal to minus 1/2 right so that's the other zero of this polynomial right so using these two we found the X square minus 3 and we divided it by that and we easily got the other zero of this polynomial ready for our next question if the polynomial X to the power 4 plus 2x cubed plus 8x squared plus 12x plus 18 is divided by another polynomial X square plus 5 the remainder comes out to be of the form px plus Q find the values of P and Q so how do you solve this question it's pretty simple you just need to divide this polynomial with this one so let's write it down so this is going to be our divisor right x squared plus 5 and we need to divide this long polynomial X to the power 4 so just write it down plus 2 x cubed plus 8x squared plus 12x plus 18 and so let me extend this line right up to here okay and so you need to sit and do this division right it's simple algebra so you need to divide it so I'm not going to do it here so I'll write down what's the quotient going to turn out to be X square plus 2x plus 3 so that comes out to be the quotient and when you divide this we will get the remainder as 2x plus 3 okay and note that the question talks only about the remainder not the quotient so take care to compare just the remainder with the form px plus Q so this basically is going to be px plus Q now it's pretty simple you just need to compare the coefficients so P is going to be 2 right just comparing the remainder with this form so we have PS - and what will Q be very simple Q is going to be 3 okay so that's our answer here let's look at our next question find the zeros of x squared plus X minus P into P plus 1 okay so let's write down our polynomial so our polynomial FX is x squared plus X minus P into P plus 1 right okay and remember to find the zeros of the polynomial we need to equate this polynomial to zero so that's our equation and we need to solve for X here okay so it's pretty simple and first let's expand it out here so X square plus X and let's expand out these terms so we are going to get minus P Square minus P equal to 0 right so now we need to factorize this to find the zeros of this polynomial so let's take the square terms together so it's X square minus P squared plus X minus P so we are doing this to factorize it okay and can you see that form a square minus B Square right so let's write that as X plus P into X minus P right so that's that and then we have plus X minus P equal to 0 right so let's take X minus P common here so what are we going to get into X plus P plus 1 okay right so either this is 0 or this is 0 right so those are our zeros of X so we can have so what do we have here so X minus P is 0 so that's X equal to P right that's one of our zeros and let's also equate this to 0 so X plus P plus 1 equals 0 which basically means X equal to minus of P plus 1 right it goes to the other side so here we found the two zeros of this polynomial it's P and minus P plus 1 that's our answer of this question and here's our final question if the product of the zeros of the polynomial ax squared minus 6x minus 6 is 4 find the value of a now I'm not going to be solving this question because I want you to try it yourself and do let me know your answer by putting it in the comments below I look forward to reading your answers so I hope the concept of polynomials is crystal clear to you now and do remember to Like comment and share out this video and if you haven't subscribed to my YouTube channel already hit the subscribe button right now and do click the notification bell to get notified about new videos you can check my Facebook page and do check out my website Manoah academy.com for more courses and questions i'll put the links below thanks for watching