Lec 02 | HSSC - l | Chap 05 | Polynomials | Topic: Applications of Polynomials | Mathematics

Added: 2026-05-23

[00:00:02]Dear Students Assalamu Alaikum Students Today We Will Discuss Some More MCQs Related To Chapter Polynomial In The Previous Lecture We Have Studied About Factor Theorem Reminder Theorem And Zeros Of The Polynomial. Now we will discuss the practical application of this concept.

[00:00:19]So start from the first MCQ.

[00:00:24]This is the statement of first MCQ.

[00:00:27]Which is the which of the following is the continuum when this polynomial is divided by x - 2 Students this is a very basic MCQ of ours which is related to the previous topic. The only difference here is that the degree of the polynomial is four. So what do you want? Need content. What you can do for that is you can do long division as well as synthetic division.

[00:00:54]Synthetic devices are easier for us.

[00:00:56]We divide synthetically. This polynomial will be written as coefficients. As explained to you earlier. 1 -5 6 -4 and 8 and we are dividing this synthetically.

[00:01:11]From which number? From here you see what is the value of a we have? Two.

[00:01:16]We students will divide it synthetically by two. Here it comes too. It's done here -3.

[00:01:22]Then we multiply two by this -3.

[00:01:24]You know the procedure. Then it became 0 - 4 and finally it became -8. How much reminder is coming for this? 0 is coming. Because what is x - 2? This is also a factor. So there could have been an MCQ related to that also.

[00:01:39]We will make a quotient from the remaining coefficients.

[00:01:42]What quotient are students becoming?

[00:01:44]Here x - 3x² + 0x - 4, it is not necessary to write the term containing 0x.

[00:01:52]x - 3x² - 4 So students, you see which of our options a b d is correct.

[00:02:04]Students, here option number D is the correct option.

[00:02:09]Now we will move to the next MCQ.

[00:02:13]If fx = x to the power 4 - 3x + 7x² - x + 5 leaves the remainder of 10 when divided by x - 1 what is f1? So you know that f1 will be called reminder here. What do you have to do? You have to take the value from here inside this fx, this value is one and you have to plus it inside this fx. End of after that you will get the reminder and how much of that reminder will we have? That reminder will be the same as the reminder given here, we have 10. So what was the condition, you might remember what f of a was equal to? r. k. The f1 here will obviously be equal to r because we are putting one in place of a and as a reminder, the given here is 10, so this means that f of one is equal to 10. Option number B students, this is the correct option here.

[00:03:12]What is the remainder when this polynomial is divided by x + 1? Students, you can find the remainder. Plug the value of f into the -1 polynomial and solve it.

[00:03:27]You will get the remainder which is -21. Option A is the correct option.

[00:03:33]Next MCQ is which of the following is not a factor of this polynomial. Here we have to see which polynomial is this and which of these factors given to us is not a factor of this polynomial. So you can check it by plugging the value in that also.

[00:03:48]What is the value of x from here? One that you put into your polynomial.

[00:03:53]You see zero is coming. So this means this is a factor. Second, you take the value from here. What value will x get?

[00:04:00]Two. You put that in your polynomial.

[00:04:03]You see this also came to zero. Meaning this is also a factor. What value are students getting from here to us? Three. When we put that inside the polynomial, that's also a factor. And the question is, what can we write for the fourth option of not a factor? x -1 From here we will take the value -1. And when you plug it into the polynomial, you see that it's not coming out equal to 0. This means that option number D is not a factor of this given polynomial. So here the correct option student we have option number D.

[00:04:41]If 2 is a zero of the polynomial. This is the polynomial, now basically this is the practical application of polynomials. Here we will use the concept of factor theorem and reminder theorem of zeros. Now you see here this two if two is the zero of the polynomial. That means if I take this p of x.

[00:05:02]This polynomial x + 5x² - 4x + k is the student p of x we ​​have.

[00:05:12]And two right here is the zero of this polynomial. That means if I subtract p from 2, what will the output be equal to? It should be equal to zero. But now it will not be equal to zero. Because what are we supposed to do? Here the unknown value K has to be found out. The polynomial is not complete. There is some unknown inside the polynomial whose value we have to find out.

[00:05:33]Here students you will calculate p of two.

[00:05:38]We're going to substitute two for x and then write this by putting p of 2 as zero because two is the zero of the polynomial. Students, we will solve this a little bit.

[00:05:52]5 * 4 = 20 - 8 + k Both this 8 and this 8 are cut off. -20 you brought here equal to k. So the value of K student we have is -20. You see that here option number B is our correct option.

[00:06:12]If 2x + 3px² - 4. Here obviously we have to calculate the value of p. Have to find it out. We have to use the concept or the reminder theorem or the factor theorem, whatever condition is given to us. Now here students, we have been reminded that there will be reminder four. If divided by x + 2. The divisor is x + 2. So what will you do? What value of x are you getting? -2 ok? You find out p of -2.

[00:06:42]Substitute -2 inside the polynomial.

[00:06:45]+ 3p -2² -4 plus -2 and this p of -2 is either zero or equal to the reminder.

[00:06:57]When does it become zero? When you're given that the value of x is a zero of the polynomial or a factor given.

[00:07:04]Here we do not have factor given because he has mentioned the reminder.

[00:07:08]So p of -2 here is equal to reminder four.

[00:07:10]Rest you will solve it and then we will get the value of p.

[00:07:20]4 = -16 + 12p + 8 4 = -16 + 8 - 8 + 12p This becomes 4 + 8 = 12p Students you can see that 12p is equal to 12 and by dividing p by 12 we are getting the value of p as one.

[00:07:44]So here option number C student is our correct option. Let's move to the next MCQ. Now this is a little bit more calculation inside it.

[00:07:55]Look here, we have been given this factor but this is a quadratic factor. Is a factor of this polynomial? And here a and b are unknown. The question here is what is the value of a?

[00:08:09]We need to find out the value of a. The values ​​of both a and b could have been found out but that would have been very lengthy.

[00:08:15]That question may come.

[00:08:18]If there is an unknown related to the polynomial, you may get an MCQ to find one or two identical values.

[00:08:26]If you have to find out the second or third value, it becomes very lengthy. That question will come to us.

[00:08:29]Ok? Now here it has asked us the value of a.

[00:08:31]In the same question he could have also asked the value of b.

[00:08:33]And what is the divisor that he has given us, which is also a factor? Well, that's a quadratic polynomial. You will be factoring this quadratic polynomial. When you do the midterm breaking down of x² - 5x + 6, we will get these two factors, students.

[00:08:53]Both of these are linear factors. Now use these factors as a factor theorem and plug them into the polynomial to get the two equations. For example, I'll use x - 2 first.

[00:09:07]This is p of x plus student x - a plus x² + bx - 12. First I'm going to plug two into this. This becomes 2² + b * 2 - 12 with 2 - a being 2² + b * 2 - 12. P of 2 is obviously zero because it told us x - 2, which is where we extracted the polynomial that was a factor of this main polynomial. Ok?

[00:09:44]How much will this cost? 8 4a + 2b - 12 This becomes students bring -4a + 2b - 4a over here. 4a - 2b = -4 2 Let's take it as common. We have created equation number one, 2a - b = -2.

[00:10:07]Similarly, now we use its next part which is the factor. Now let's find P of 3. What is this going to be for P of 3? 3 with a 3² + b with 3 and -1 3 of 3 will also be written as zero. 27 - 9a + 3b - 12 0 = 27 - 12 15 - 9a + 3b We bring 9a and 3b here.

[00:10:46]- 3b = 15 From here students we can take three common ones. 3a - b = 5 This becomes equation number two. Now if we subtract these two. Look, there is -b here too.

[00:10:59]There is also - b here. These two equations can be subtracted and the substitution method can be used to find the value of a. So I'm starting to subtract the equations here. I started subtracting equation two from equation one. 2a - b = -2 and 3a - b = 5 perform subtraction. You know the signs will change.

[00:11:31]See more This b cancelled out. 2a - 3a -a -2 and -5 -7 This means that if the negative sign is cancelled from both the sides then the value of a is seven, which is option number C. So there will be some MCQs in which we will have to solve a little more and maximum MCQs will be such which will be just concept based and there are some MCQs in which we have to do a little working but we will also have to face one or two MCQs in which there is a little more working. Next we move to the next MCQ. Polynomial p of x leaves remainder 3 1 / x - 2 and remainder 7 1 / x - 3 Now what you need to do here is what you need to do is p of two leaves three because when you're dividing by x - 2, reminder, we have three. Ok? And when did reminder seven happen? When we divide by x - 3, that is, what should be the result of p of 3? So now what you do is try putting in these four options two and three and see.

[00:12:48]In the first option, when you put three or two, see, we are getting two which we did not want. p off two we needed equal to three. If we take out P of 2 in option number C, then you can put that also in option B. Not there. If you put inside option number C then no. See in option D, when you are taking out p of two then three is coming and when you are taking out p of 3 then it is coming from.

[00:13:10]So this means that here this option number D is our correct option.

[00:13:17]Next MCQ is Factors of -2 - x + x², you arrange it and write it down. x² - x - 2 and then just factorize with the help of the midterm breaking method. So you will see that option number D will be the correct option here. From the students' point of view, this MCQ was a company models. Now these are some practical applications.

[00:13:44]Two-three MCQs related to daily life are related to this. A company model daily profit in ₹1000 by P of X. This is P of X given to us which is telling us the daily profit of a company.

[00:13:57]Where X is the number of products sold. So what does p of 10 represent? This means that X here means 10 in place of X and X represents the number of products. So you see option number B is written that daily profit when 10 products are sold. Option number B, you see, this is the correct option.

[00:14:19]See Next MCQ Student. Area of ​​the Rectangular Garden. This gives us the area of ​​a rectangular garden from a of x, which is also factorized.

[00:14:28]What do the factors x + 3 and x + 4 represent? So do you know what the students' area is? Length multiplied by width. So who are these factors here representing? To the length and breadth of the garden. Option number D is our correct option here.

[00:14:48]A car manufacturer uses C of this is a polynomial to estimate the fuel consumption for different engine settings. What is the main purpose of using polynomial model here. If you see, the option to decorate the graph is obviously not correct. To calculate only integers. There is no question of integers here. To avoid measurements. It is not like that either. What we are doing here is relating it to real life, that is, we represent anything in such a way that we can relate it to real life.

[00:15:22]So the polynomial here is designed to predict real life performance values.

[00:15:27]Next MCQ is The Volume of an Open Box. This is the volume.

[00:15:33]is a polynomial of the volume. Wear x size of squares cut from corners. What does the polynomial help with? Determine. Colour of the box is obviously not ok Weight of the cardboard is obviously not Number of boxes sold This also does not have any sole sale purchase There is no problem here Maximum possible volume Volume of the box Here students, we have a representation of the volume of the box. Now this could have been the maximum possible volume or the minimum possible volume minimum. It depends on what values ​​are coming to us and this is a new concept in mathematics which you will study in further classes.

[00:16:13]But here the volume of the box is being represented by the function of v of x.

[00:16:20]Now, students, this MCQ is related to the previous topic. The only purpose of including it here was that this MCQ was there in the recent board paper of 2026 which was held a few days ago. It is a very simple MCQ.

[00:16:34]In polynomial divis the degree of reminder is. So it was also discussed in the previous lecture that the degree of the remainder is always less than the divisor.

[00:16:43]So see here the option of less than divisor is correct. Ok? And the next MCQ was also in the board exam paper of 2026. If x - 2 is a factor of fx then f2 we need to find f2. You see, the word factor is being used here. This means that f2 is equal to what? Reminder K. And the remainder becomes zero if we are dividing by a factor.

[00:17:06]So f of two over here is going to be zero.

[00:17:08]Option number B is the students correct option. So these were some MCQs that can be made related to this chapter. Here you have its quick review reference.

[00:17:20]You should revise it and discuss it with your teachers in your classes.

[00:17:25]Thank you so much students.