CBSE • Class 10 • Mathematics • Chapter 2 (Polynomials) • Exercise 2.2

Exercise 2.2: Polynomials — NCERT Solutions

Relationship between zeros and coefficients of a quadratic polynomial.

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What this exercise covers

Sum of zerosProduct of zerosConstructing a quadratic from given zeros

Step-by-step solutions — Exercise 2.2

7 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 2.2 Q1 • 2 marks

Find the zeros of the quadratic polynomial x2−7x+10x^2 - 7x + 10 and verify the relationship between the zeros and the coefficients.
Hint (Socratic — try this first)▾
Which two numbers multiply to 1010 and add to 77?
Step-by-step solution▾

Factorise: x2−7x+10=x2−5x−2x+10=x(x−5)−2(x−5)=(x−5)(x−2)x^2 - 7x + 10 = x^2 - 5x - 2x + 10 = x(x-5) - 2(x-5) = (x-5)(x-2).

Zeros: x=5x = 5 and x=2x = 2.

Verification (for ax2+bx+cax^2 + bx + c, here a=1,b=−7,c=10a=1, b=-7, c=10):

  • Sum of zeros =5+2=7= 5 + 2 = 7 and −ba=−−71=7-\dfrac{b}{a} = -\dfrac{-7}{1} = 7. ✓
  • Product of zeros =5×2=10= 5 \times 2 = 10 and ca=101=10\dfrac{c}{a} = \dfrac{10}{1} = 10. ✓

The relationships hold.

Common mistake:
Writing sum of zeros as ba\frac{b}{a} instead of −ba-\frac{b}{a} (forgetting the negative sign).
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Exercise 2.2 Q2 • 2 marks

Find the zeros of 6x2−7x−36x^2 - 7x - 3 and verify the relationship between the zeros and its coefficients.
Hint (Socratic — try this first)▾
Split the middle term using two numbers whose product is 6×(−3)6 \times (-3) and sum is −7-7.
Step-by-step solution▾

Here a=6,b=−7,c=−3a=6, b=-7, c=-3, product ac=−18ac = -18. We need two numbers with product −18-18 and sum −7-7: these are −9-9 and 22.

6x2−7x−3=6x2−9x+2x−3=3x(2x−3)+1(2x−3)=(2x−3)(3x+1)6x^2 - 7x - 3 = 6x^2 - 9x + 2x - 3 = 3x(2x-3) + 1(2x-3) = (2x-3)(3x+1).

Zeros: 2x−3=0⇒x=322x - 3 = 0 \Rightarrow x = \dfrac{3}{2}; 3x+1=0⇒x=−133x + 1 = 0 \Rightarrow x = -\dfrac{1}{3}.

Verification:

  • Sum =32−13=9−26=76= \dfrac{3}{2} - \dfrac{1}{3} = \dfrac{9-2}{6} = \dfrac{7}{6} and −ba=−−76=76-\dfrac{b}{a} = -\dfrac{-7}{6} = \dfrac{7}{6}. ✓
  • Product =32×(−13)=−12= \dfrac{3}{2} \times \left(-\dfrac{1}{3}\right) = -\dfrac{1}{2} and ca=−36=−12\dfrac{c}{a} = \dfrac{-3}{6} = -\dfrac{1}{2}. ✓
Common mistake:
Splitting the middle term using numbers whose product is cc instead of acac when the leading coefficient is not 1.
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Exercise 2.2 Q3 • 2 marks

Find a quadratic polynomial whose sum of zeros is −3-3 and product of zeros is 22.
Hint (Socratic — try this first)▾
How can you build a quadratic directly from the sum SS and product PP of its zeros?
Step-by-step solution▾

A quadratic polynomial with sum of zeros SS and product of zeros PP can be written as: k[x2−Sx+P]k\left[x^2 - Sx + P\right]

Here S=−3S = -3 and P=2P = 2: x2−(−3)x+2=x2+3x+2.x^2 - (-3)x + 2 = x^2 + 3x + 2.

Taking k=1k = 1, the required polynomial is x2+3x+2\boxed{x^2 + 3x + 2}.

(Any non-zero multiple, e.g. 2x2+6x+42x^2 + 6x + 4, is also valid.)

Common mistake:
Writing x2+Sx+Px^2 + Sx + P instead of x2−Sx+Px^2 - Sx + P, i.e. using the wrong sign for the sum term.
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Exercise 2.2 Q4 • 2 marks

Find a quadratic polynomial whose zeros are 14\dfrac{1}{4} and −1-1.
Hint (Socratic — try this first)▾
First compute the sum and product of the two given zeros.
Step-by-step solution▾

Let the zeros be α=14\alpha = \dfrac{1}{4} and β=−1\beta = -1.

  • Sum S=14+(−1)=−34S = \dfrac{1}{4} + (-1) = -\dfrac{3}{4}.
  • Product P=14×(−1)=−14P = \dfrac{1}{4} \times (-1) = -\dfrac{1}{4}.

Polynomial =x2−Sx+P=x2+34x−14= x^2 - Sx + P = x^2 + \dfrac{3}{4}x - \dfrac{1}{4}.

Multiplying by 44 to clear fractions: 4x2+3x−1.\boxed{4x^2 + 3x - 1}.

Common mistake:
Forgetting to multiply through to remove fractions, or multiplying incorrectly so the zeros change.
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Exercise 2.2 Q5 • 2 marks

If α\alpha and β\beta are the zeros of x2−5x+6x^2 - 5x + 6, find the value of 1α+1β\dfrac{1}{\alpha} + \dfrac{1}{\beta}.
Hint (Socratic — try this first)▾
Can you express 1α+1β\frac{1}{\alpha}+\frac{1}{\beta} using only the sum and product of the zeros?
Step-by-step solution▾

For x2−5x+6x^2 - 5x + 6 (a=1,b=−5,c=6a=1, b=-5, c=6):

  • α+β=−ba=5\alpha + \beta = -\dfrac{b}{a} = 5
  • αβ=ca=6\alpha\beta = \dfrac{c}{a} = 6

Now combine the fractions: 1α+1β=β+ααβ=56.\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} = \frac{5}{6}.

So the value is 56\dfrac{5}{6}.

Common mistake:
Trying to find α\alpha and β\beta separately and then adding reciprocals, which is longer and error-prone; using the identity is quicker.
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Exercise 2.2 Q6 • 3 marks

If one zero of the polynomial 2x2+kx+62x^2 + kx + 6 is 22, find the value of kk and the other zero.
Hint (Socratic — try this first)▾
What must be true about p(2)p(2) if 22 is a zero?
Step-by-step solution▾

Since 22 is a zero, p(2)=0p(2) = 0: 2(2)2+k(2)+6=0⇒8+2k+6=0⇒2k=−14⇒k=−7.2(2)^2 + k(2) + 6 = 0 \Rightarrow 8 + 2k + 6 = 0 \Rightarrow 2k = -14 \Rightarrow k = -7.

So the polynomial is 2x2−7x+62x^2 - 7x + 6.

Other zero: Product of zeros =ca=62=3= \dfrac{c}{a} = \dfrac{6}{2} = 3.

If one zero is 22 and the other is β\beta, then 2β=3⇒β=322\beta = 3 \Rightarrow \beta = \dfrac{3}{2}.

Therefore k=−7k = -7 and the other zero is 32\dfrac{3}{2}.

Common mistake:
Using product of zeros as cc instead of ca\frac{c}{a}, giving the wrong second zero.
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Exercise 2.2 Q7 • 3 marks

If α\alpha and β\beta are the zeros of the quadratic polynomial x2−6x+8x^2 - 6x + 8, find the value of α2+β2\alpha^2 + \beta^2.
Hint (Socratic — try this first)▾
Which identity connects α2+β2\alpha^2+\beta^2 with (α+β)(\alpha+\beta) and αβ\alpha\beta?
Step-by-step solution▾

For x2−6x+8x^2 - 6x + 8:

  • α+β=−−61=6\alpha + \beta = -\dfrac{-6}{1} = 6
  • αβ=81=8\alpha\beta = \dfrac{8}{1} = 8

Use the identity α2+β2=(α+β)2−2αβ\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta: α2+β2=62−2(8)=36−16=20.\alpha^2 + \beta^2 = 6^2 - 2(8) = 36 - 16 = 20.

So α2+β2=20\alpha^2 + \beta^2 = 20.

Common mistake:
Writing α2+β2=(α+β)2\alpha^2 + \beta^2 = (\alpha+\beta)^2 and forgetting to subtract 2αβ2\alpha\beta.
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How to approach Exercise 2.2

  1. Re-read the chapter summary first. Open Polynomials and refresh the key concepts: Zeros of a polynomial, Sum and product of zeros, Quadratic polynomials, Cubic polynomials.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Polynomials

  1. Exercise 2.1 — Geometric meaning of zeros — reading zeros of a polynomial off its graph.
  2. Exercise 2.2 — Relationship between zeros and coefficients of a quadratic polynomial.

← Exercise 2.1

Geometric meaning of zeros — reading zeros of a polynomial off its graph.

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