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Exercise 2.1 (NCERT) - Polynomials, Class 10

Chapter 2 Exercise 2.1 Class 10

Question 1

The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

The number of zeros of a graph is the number of times the graph intersects the x-axis. 

The answers are written on the graphs below in red circle.

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Exercise 2.2 (NCERT) - Polynomials, Class 10

Chapter 2 Exercise 2.2 Class 10

Below are the Quick links for all questions of Exercise 2.2, Real Numbers, Class 10.

Click a link to view the solution of corresponding question.

Relationships of Zeros and Coefficients

Splitting the Middle Term

Relationships of Zeros and Coefficients

This exercise is about very basic concepts of Zeros of Polynomials. The zero of a polynomial, P(x), is the value of x for which the value of P(x) is 0. For a quadratic polynomial ax2+bx+c there can be exactly 2 real zeros, say α and β. The relationships between the zeros of the quadratic polynomial and the coefficients of the polynomial are:

𝑃(𝑥)=𝑎𝑥2+𝑏𝑥+𝑐𝑆=Sum of zeros=α+β=−𝑏𝑎𝑃=Product of zeros=α×β=𝑐𝑎

Now if we already know the Sum (S) and Product (P) of a polynomial, the we can directly get the polynomial using the formula written below.

𝑆=Sum of zeros𝑃=Product of zeros𝑡ℎ𝑒𝑛  𝑃(𝑥)=𝑥2−𝑆𝑥+𝑃

Splitting the Middle Term

If you are able to understand the image displayed below then you will solve all questions really easily.

Before we start solving the questions of this exercise, it is very important to understand the concept used to solve these questions – Splitting the Middle Term of a quadratic polynomial. You are suggested to watch the video embedded below completely to understand 🧠 the concept. The tutorial is slightly lengthy so please have some patience in case you want to dive deep into it. 🤿 👀 ❤️ 💃

How it works?

The coefficient of x, b, is split in two parts, say b1 and b2, such that

b1 + b2 = b and

b1 x b2 = a x c

To find the required pair of b1 and b2 you must use prime factorisation of b. 

Do you love Mathematics? If yes, then watch this video to dive deep into the underlying concept.

Below are the solutions of every question of exercise 2.2 of chapter 1, Polynomials, NCERT of class 10. YouTube video tutorial of every question is also embedded along with the written solution.  

Question 1

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) \( \begin{aligned} x^2-2x-8 \end{aligned}\)

𝑥2−2𝑥−8=𝑥2−4𝑥+2𝑥−8=𝑥(𝑥−4)+2(𝑥−4)=(𝑥−4)(𝑥+2)
𝑃𝑢𝑡𝑡𝑖𝑛𝑔  (𝑥−4)(𝑥+2)=0
⇒  𝑥−4=0  ⇒  𝑥=4  𝑜𝑟  𝑥+2=0𝑜𝑟  𝑥=−2

Answer: 4 & -2 are the zeros of the polynomial.

Verification

Comparing the polynomial with ax2 + bx + c

𝑎=1,𝑏=−2,𝑐=−8Sum of roots=−𝑏𝑎=−(−2)1=2Sum of roots=α+β=4+(−2)=2𝐻𝑒𝑛𝑐𝑒  α+β=−𝑏𝑎
Product of roots=𝑐𝑎=−81=−8Product of roots=α×β=4×(−2)=−8𝐻𝑒𝑛𝑐𝑒  α×β=𝑐𝑎

(ii) \( \begin{aligned} 4s^2-4s + 1 \end{aligned}\)

4𝑠2−4𝑠+1=4𝑠2−2𝑠−2𝑠+1=2𝑠(2𝑠−1)−1(2𝑠−1)=(2𝑠−1)(2𝑠−1)
𝑃𝑢𝑡𝑡𝑖𝑛𝑔  (2𝑠−1)(2𝑠−1)=0
⇒2𝑠−1=0  𝑜𝑟  2𝑠−1=0⇒𝑠=12  𝑜𝑟  𝑠=12

Answer: 1/2 & 1/2 are the zeros of the polynomial.

Verification

Comparing the polynomial with ax2 + bx + c

𝑎=4,𝑏=−4,𝑐=1Sum of roots=−𝑏𝑎=−(−4)4=1Sum of roots=α+β=12+12=1𝐻𝑒𝑛𝑐𝑒  α+β=−𝑏𝑎
Product of roots=𝑐𝑎=14Product of roots=α×β=12×12=14𝐻𝑒𝑛𝑐𝑒  α×β=𝑐𝑎

(iii) \( \begin{aligned} 6x^2-3-7x \end{aligned}\)

6𝑥2−3−7𝑥=6𝑥2−7𝑥−3=6𝑥2−9𝑥+2𝑥−3=3𝑥(2𝑥−3)+1(2𝑥−3)=(2𝑥−3)(3𝑥+1)
𝑃𝑢𝑡𝑡𝑖𝑛𝑔  (2𝑥−3)(3𝑥+1)=0
⇒2𝑥−3=0  ⇒𝑥=32  𝑜𝑟  3𝑥+1=0𝑜𝑟  𝑥=−13

Answer: 3/2 & -1/3 are the zeros of the polynomial.

Verification

Comparing the polynomial with ax2 + bx + c

𝑎=6,𝑏=−7,𝑐=−3Sum of roots=−𝑏𝑎=−(−7)6=76Sum of roots=α+β=32–13=9−26=76𝐻𝑒𝑛𝑐𝑒  α+β=−𝑏𝑎
Product of roots=𝑐𝑎=−36=−12Product of roots=α×β=32×−13=−12𝐻𝑒𝑛𝑐𝑒  α×β=𝑐𝑎

(iv) \( \begin{aligned} 4u^2+ 8u \end{aligned}\)

4𝑢2+8𝑢=4𝑢(𝑢+2)
𝑃𝑢𝑡𝑡𝑖𝑛𝑔  4𝑢(𝑢+2)=0
⇒4𝑢=0  ⇒𝑢=0  𝑜𝑟  𝑢+2=0𝑜𝑟  𝑢=−2

Answer: 0 & -2 are the zeros of the polynomial.

Verification

Comparing the polynomial with ax2 + bx + c

𝑎=4,𝑏=8,𝑐=0Sum of roots=−𝑏𝑎=−84=−2Sum of roots=α+β=0+(−2)=−2𝐻𝑒𝑛𝑐𝑒  α+β=−𝑏𝑎
Product of roots=𝑐𝑎=04=0Product of roots=α×β=0×(−2)=0𝐻𝑒𝑛𝑐𝑒  α×β=𝑐𝑎

(v) \( \begin{aligned} t^2-15 \end{aligned}\)

𝑡2−15=(𝑡+15‾‾‾√)(𝑡−15‾‾‾√)
𝑃𝑢𝑡𝑡𝑖𝑛𝑔  (𝑡+15‾‾‾√)(𝑡−15‾‾‾√)=0
⇒𝑡+15‾‾‾√=0  ⇒𝑡=−15‾‾‾√  𝑜𝑟  𝑡−15‾‾‾√=0𝑜𝑟  𝑡=15‾‾‾√

Answer: √15 & -√15 are the zeros of the polynomial.

Verification

Comparing the polynomial with ax2 + bx + c

𝑎=1,𝑏=0,𝑐=−15Sum of roots=−𝑏𝑎=01=0Sum of roots=α+β=15‾‾‾√–15‾‾‾√=0𝐻𝑒𝑛𝑐𝑒  α+β=−𝑏𝑎
Product of roots=𝑐𝑎=−151=−15Product of roots=α×β=15‾‾‾√×(−15‾‾‾√)=−15𝐻𝑒𝑛𝑐𝑒  α×β=𝑐𝑎

(vi) \( \begin{aligned} 3x^2-x-4 \end{aligned}\)

3𝑥2−𝑥−4=3𝑥2−4𝑥+3𝑥−4=𝑥(3𝑥−4)+1(3𝑥−4)=(3𝑥−4)(𝑥+1)
𝑃𝑢𝑡𝑡𝑖𝑛𝑔  (3𝑥−4)(𝑥+1)=0
⇒3𝑥−4=0  ⇒𝑥=43  𝑜𝑟  𝑥+1=0𝑜𝑟  𝑥=−1

Answer: 4/3 & -1 are the zeros of the polynomial.

Verification

Comparing the polynomial with ax2 + bx + c

𝑎=3,𝑏=−1,𝑐=−4Sum of roots=−𝑏𝑎=−(−1)3=13Sum of roots=α+β=43−1=13𝐻𝑒𝑛𝑐𝑒  α+β=−𝑏𝑎
Product of roots=𝑐𝑎=−43Product of roots=α×β=43×(−1)=−43𝐻𝑒𝑛𝑐𝑒  α×β=𝑐𝑎

Question 2

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i) 1/4, -1

𝑆=Sum of zeros=14𝑃=Product of zeros=−1

Now P(x) = x2 – Sx + P

𝑃(𝑥)=𝑥2−𝑆𝑥+𝑃=𝑥2−(14)𝑥+(−1)=𝑥2−𝑥4−1

We can stop solving the question further and the polynomial written above is the correct answer. However, if we multiply this polynomial by any non-zero real number, the resulting polynomial will also have the same zeros.

Multiply the P(x) by 4 to remove 4 from the denominator. 

4(𝑥2−𝑥4−1)=4𝑥2−𝑥−4

(ii) √2 , 1/3 

𝑆=Sum of zeros=2‾√𝑃=Product of zeros=13

Now P(x) = x2 – Sx + P

𝑃(𝑥)=𝑥2−𝑆𝑥+𝑃=𝑥2−2‾√𝑥+13

Multiply the P(x) by 3 to remove 3 from the denominator. 

3(𝑥2−2‾√𝑥+13)=3𝑥2−32‾√𝑥+1

(iii) 0, √5

𝑆=Sum of zeros=0𝑃=Product of zeros=5‾√

Now P(x) = x2 – Sx + P

𝑃(𝑥)=𝑥2−𝑆𝑥+𝑃=𝑥2−0𝑥+5‾√=𝑥2+5‾√

(iv) 1, 1

𝑆=Sum of zeros=1𝑃=Product of zeros=1

Now P(x) = x2 – Sx + P

𝑃(𝑥)=𝑥2−𝑆𝑥+𝑃=𝑥2−1𝑥+1=𝑥2−𝑥+1

(v) -1/4, 1/4 

𝑆=Sum of zeros=−14𝑃=Product of zeros=14

Now P(x) = x2 – Sx + P

𝑃(𝑥)=𝑥2−𝑆𝑥+𝑃=𝑥2−(−14)𝑥+14=𝑥2+14𝑥+14

Multiply the P(x) by 4 to remove 4 from the denominator. 

4(𝑥2+14𝑥+14)=4𝑥2+𝑥+1

(vi) 4, 1

𝑆=Sum of zeros=4𝑃=Product of zeros=1

Now P(x) = x2 – Sx + P

𝑃(𝑥)=𝑥2−𝑆𝑥+𝑃=𝑥2−4𝑥+1
\( \begin{aligned} \end{aligned}\)

The post Exercise 2.2 (NCERT) – Polynomials, Class 10 – All Solutions appeared first on Mathemafia.

]]> Class 10 | Chapter 2 | Polynomials | Exercise 2.1 https://mathemafia.com/class-10-chapter-2-polynomials-exercise-2-1/ Sun, 20 Jun 2021 04:41:12 +0000 http://sh016.global.temp.domains/~mathemaf/?p=7075 Question 1: The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case. Class 10 | Examples Class 10 | Exercise 2.2

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Question 1: The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Class 10 | Examples
Class 10 | Exercise 2.2

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Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

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Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

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❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

Super Cool Stuff

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]]> Class 10 | Chapter 2 | Polynomials | Exercise 2.3 https://mathemafia.com/class-10-chapter-2-polynomials-exercise-2-3/ Sun, 20 Jun 2021 03:19:37 +0000 http://sh016.global.temp.domains/~mathemaf/?p=7033 Question 1: Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following : (i) p(x) = x3 – 3×2 + 5x – 3, g(x) = x2 – 2 (ii) p(x) =…

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Question 1: Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :
 (i) p(x) = x3 – 3x2 + 5x – 3, g(x) = x2 – 2
 (ii) p(x) = x4 – 3x2 + 4x + 5, g(x) = x2 + 1 – x
 (iii) p(x) = x4 – 5x + 6, g(x) = 2 – x2

Question 2: Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: (i) t2 – 3 , 2t4 + 3t3 – 2t2 – 9t – 12
(ii) x2 + 3x + 1, 3x4 + 5x3 – 7x2 + 2x + 2 (iii) x3 – 3x + 1, x5 – 4x3 + x2 + 3x + 1

Question 3: Obtain all other zeroes of 3x4 + 6x3 – 2x2 – 10x – 5, if two of its zeroes are  5/3 and – 5/3⋅

Question 4: On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x).

Question 5: Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0

Class 10 | Exercise 2.2
Class 10 | Chapter 3 | Theorems & Concepts

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Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

 

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

– Rishi’s Parents (USA)

Rishi (USA Student)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

Also, your videos on Mathemafia youtube channel are very informative.

 

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

Super Cool Stuff

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]]> Class 10 | Chapter 2 | Polynomials | Exercise 2.2 https://mathemafia.com/class-10-chapter-2-polynomials-exercise-2-2/ Sun, 20 Jun 2021 03:19:13 +0000 http://sh016.global.temp.domains/~mathemaf/?p=7032 Question 1: Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x2 –2x–8 (ii) 4s2 –4s+1 (iii) 6x2 –3–7x (iv) 4u2 +8u (v) t2 –15 (vi) 3x2 –x–4 Question 2:…

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Question 1: Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x2 –2x–8 (ii) 4s2 –4s+1 (iii) 6x2 –3–7x (iv) 4u2 +8u (v) t2 –15 (vi) 3x2 –x–4

Question 2: Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

i) 1/4,−1 (ii) √2, 1/3 (iii) 0, √5 (iv) 1,1 (v) − 1/4, 1/4 (vi) 4,1

Class 10 | Exercise 2.1
Class 10 | Exercise 2.3

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Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

 

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

– Rishi’s Parents (USA)

Rishi (USA Student)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

Also, your videos on Mathemafia youtube channel are very informative.

 

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

Super Cool Stuff

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]]> Class 10 | Chapter 2 | Polynomials | Examples https://mathemafia.com/class-10-chapter-2-polynomials-examples/ Sun, 20 Jun 2021 03:13:40 +0000 http://sh016.global.temp.domains/~mathemaf/?p=7025 Example 1: Look at the graphs in Fig. 2.9 given below. Each is the graph of y = p(x), where p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x). Example 2: Find the…

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Example 1: Look at the graphs in Fig. 2.9 given below. Each is the graph of y = p(x), where p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x).

Example 2: Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Example 3: Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the coefficients.

Example 4: Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.

Example 5: Verify that 3, –1, −1/3 are the zeroes of the cubic polynomial p(x) = 3x3 – 5x2 – 11x – 3, and then verify the relationship between the zeroes and the coefficients.

Example 6: Divide 2x2 +3x+1 by x+2.

Example 7: Divide 3x3 +x2 +2x+5 by 1+2x+x2.

Example 8: Divide 3x2 – x3 – 3x + 5 by x – 1 – x2, and verify the division algorithm.

Example 9: Find all the zeroes of 2x4 –3x3 –3x2 +6x–2, if you know that two of its zeroes are 2 and − 2.

Class 10 | Theorems & Concepts
Class 10 | Exercise 2.1

Testimonials

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Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

 

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

– Rishi’s Parents (USA)

Rishi (USA Student)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

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❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

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