Home
NCERT Soltions
Class 10
Mathematics
Polynomials
Find the zeroes of the following quadratic polynomials and verify the ...

Find the zeroes of the following quadrat | Class 10 Mathematics Chapter Polynomials, Polynomials NCERT Solutions

Welcome to the NCERT Solutions for Class 10 Mathematics - Chapter Polynomials. This page offers a step-by-step solution to the specific question from Exercise 2, Question 1: . With detailed answers and explanations for each chapter, students can strengthen their understanding and prepare confidently for exams. Ideal for CBSE and other board students, this resource will simplify your study experience.

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

Answer:

(i) x² – 2x – 8

= x – 4x + 2x – 8

= x(x – 4) + 2(x – 4)

= (x + 2) (x – 4)

The value of x² – 2x – 8 is zero if (x + 2) = 0 and (x – 4) = 0

x = -2 or x = 4

Sum of zeroes = (-2 + 4) = 2 = - coefficient of x

coefficient of x²

Product of zeroes = (-2) × 4 = -8 = Constant term

coefficient of x²

(ii) 4s²– 4s + 1

= 4s²– 2s – 2s + 1

= 2s (2s – 1) – 1 (2s – 1)

= ( 2s – 1 ) ( 2s – 1 )

The value of 4s² – 4s + 1 is zero , if (2s-1) = 0 and (2s-1 ) = 0

s = 1/2 , 1/2

Sum of zeroes = (1/2 + 1/2) = 1 - coefficient of x

coefficient of x²

Product of zeroes =1/2 × 1/2 = 1/4 = constant term

coefficient of x²

(iii) 6x²–7x – 3

= 6x – 9x + 2x – 3

= 3x (2x – 3) + 1(2x – 3)

= (3x + 1) (2x – 3)

The value of 6x²–7x – 3 is zero, if (3x + 1) = 0 and (2x – 3) = 0

X = -1 /3 , 3/2

Sum of zeroes = ( -1/3 + 3/2) = 7/6 = - coefficient of x

coefficient of x²

Product of zeroes = -1/3 × 3/2 = -3/2 = constant term

coefficient of x²

(iv) 4u²+8u

4u(u+2)

The value of 4u²+8u is zero, if 4u = 0 and (u+2) =0

u = 0, - 2

Sum of zeroes = ( 0+ (-2)) = -2 = - coefficient of x

coefficient of x²

Product of zeroes = (-2) × 0 = 0 = constant term

coefficient of x²

(v)

solution

(vi)

3x²–x–4

3x – 4x + 3x – 4

= x (3x – 4) + 1 (3x – 4)

The value of 3x – x + 4 is zero, if (3x – 4) = 0 and (x + 1) = 0

Sum of zeroes = [4/3 + ( -1)] = 1/3 = - coefficient of x

coefficient of x²

Product of zeroes = (-1) × 4/3 = -4/3 = constant term

coefficient of x²

Other Questions of Chapter Polynomials

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

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

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

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

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

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.

If the zeroes of the polynomial x³– 3x²+ x + 1 are a – b, a, a + b, find a and b.

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

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

Popular questions of class 10 Mathematics

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x²– 3x + 5 = 0 (iii) 2x²– 6x + 3 = 0

Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

Find two consecutive positive integers, sum of whose squares is 365.

Find two numbers whose sum is 27 and product is 182.

Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i) intersecting lines (ii) parallel lines (iii) coincident lines

Refer to Example 13. (i) Complete the following table:

(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability

Do you agree with this argument? Justify your answer.

Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.

A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Recently Viewed Questions of Class 10 Mathematics

Suppose you drop a die at random on the rectangular region shown in Fig. 15.6. What is the probability that it will land inside the circle with diameter 1m?

Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to
(A) 50° (B) 60°
(C) 70° (D) 80°

A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3.

Find the number of blue balls in the jar.

Find two consecutive positive integers, sum of whose squares is 365.

Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Study Tips for Answering NCERT Questions:

NCERT questions are designed to test your understanding of the concepts and theories discussed in the chapter. Here are some tips to help you answer NCERT questions effectively:

Read the question carefully and focus on the core concept being asked.
Reference examples and data from the chapter when answering questions about Polynomials.
Review previous year question papers to get an idea of how such questions may be framed in exams.
Practice answering questions within the time limit to improve your speed and accuracy.
Discuss your answers with your teachers or peers to get feedback and improve your understanding.

Latest Blog Posts

Stay updated with our latest educational content and study tips

The Secret to Smarter Learning — Building Strong Critical Thinking Skills

In today’s world of endless information , knowing how to think is more important than knowing what to think . From school projects to real – life decisions , critical thinking helps students question ideas , analyze facts and form logical conclusions . But what exactly does critical thinking mean ? Simply put , it’s […]

Study Smarter, Not Harder: Build Productive Habits That Stick

Every student dreams of better grades , stronger focus and more study time – but the real challenge isn’t starting, it’s staying consistent . Building productive study habits is not about studying all day , it’s about studying smart . In today’s fast – paced digital world, distractions are everywhere – from endless phone notifications […]

The Hidden Risks of Online Gaming for Children — Is your child safe while gaming online?

Online gaming has rapidly become one of the most popular pastimes among children. Whether it’s multiplayer mobile games , PC adventures or console challenges , kids are spending more time than ever in the virtual world . On the surface, gaming seems entertaining and even educational – improving hand- eye coordination , teamwork and problem […]

The Role of Parents in Digital Literacy – Guiding Kids for a Smarter Online Future

Kids today are surrounded by screens from the moment they wake up . Whether it’s smart classrooms, online lessons or video games with friends technology has quietly become a part of everything they do. It’s amazing how much they can learn, explore and create with just a tap or a click. But it also brings […]

View All Blog Posts