Exercise 2.2 answers: Polynomials

Class 10 Maths2 questions

Exercise 2.2

2 questions · page 23 of the book

Question 1

“Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.” · p. 23

Open NCERT p. 23Matches NCERT’s answer

(i) x² − 2x − 8

  1. Split the middle term: −2x = −4x + 2x, so x² − 4x + 2x − 8.
  2. Group and factorise: x(x − 4) + 2(x − 4) = (x − 4)(x + 2).
  3. Zeroes: x = 4 and x = −2.
  4. Sum of zeroes = 4 + (−2) = 2. Compare with −b/a = −(−2)/1 = 2. They match.
  5. Product of zeroes = 4 × (−2) = −8. Compare with c/a = −8/1 = −8. They match.

AnswerZeroes are 4 and −2.

(ii) 4s² − 4s + 1

  1. Split the middle term: −4s = −2s − 2s, so 4s² − 2s − 2s + 1.
  2. Group and factorise: 2s(2s − 1) − 1(2s − 1) = (2s − 1)(2s − 1).
  3. Zero: s = 1/2, repeated (the same value both times).
  4. Sum of zeroes = 1/2 + 1/2 = 1. Compare with −b/a = −(−4)/4 = 1. They match.
  5. Product of zeroes = 1/2 × 1/2 = 1/4. Compare with c/a = 1/4. They match.

AnswerZero is 1/2 (repeated).

(iii) 6x² − 3 − 7x

  1. Read the polynomial in order first: 6x² − 3 − 7x is the same as 6x² − 7x − 3.
  2. Split the middle term: −7x = −9x + 2x, so 6x² − 9x + 2x − 3.
  3. Group and factorise: 3x(2x − 3) + 1(2x − 3) = (2x − 3)(3x + 1).
  4. Zeroes: x = 3/2 and x = −1/3.
  5. Sum of zeroes = 3/2 + (−1/3) = 7/6. Compare with −b/a = −(−7)/6 = 7/6. They match.
  6. Product of zeroes = (3/2) × (−1/3) = −1/2. Compare with c/a = −3/6 = −1/2. They match.

AnswerZeroes are 3/2 and −1/3.

(iv) 4u² + 8u

  1. Factorise by taking 4u common: 4u² + 8u = 4u(u + 2).
  2. Zeroes: u = 0 and u = −2.
  3. Sum of zeroes = 0 + (−2) = −2. Compare with −b/a = −8/4 = −2. They match.
  4. Product of zeroes = 0 × (−2) = 0. Compare with c/a = 0/4 = 0. They match.

AnswerZeroes are 0 and −2.

(v) t² − 15

  1. t² − 15 = 0 means t² = 15, so t = √15 or t = −√15.
  2. Sum of zeroes = √15 + (−√15) = 0. Compare with −b/a = −0/1 = 0. They match.
  3. Product of zeroes = √15 × (−√15) = −15. Compare with c/a = −15/1 = −15. They match.

AnswerZeroes are √15 and −√15.

(vi) 3x² − x − 4

  1. Split the middle term: −x = −4x + 3x, so 3x² − 4x + 3x − 4.
  2. Group and factorise: x(3x − 4) + 1(3x − 4) = (3x − 4)(x + 1).
  3. Zeroes: x = 4/3 and x = −1.
  4. Sum of zeroes = 4/3 + (−1) = 1/3. Compare with −b/a = −(−1)/3 = 1/3. They match.
  5. Product of zeroes = (4/3) × (−1) = −4/3. Compare with c/a = −4/3. They match.

AnswerZeroes are 4/3 and −1.

Watch this explained “Two numbers hiding in the coefficients”, 0:00 into What the sum and product of two zeroes reveal about a, b and c

Question 2

“Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.” · p. 23

Open NCERT p. 23Checked by computerAnswers can differ: one example

(i) 1/4, −1

  1. S = 1/4 and P = −1.
  2. x² − (1/4)x + (−1) = x² − x/4 − 1.
  3. Check: −b/a = 1/4 and c/a = −1.

Answerx² − x/4 − 1

(ii) √2, 1/3

  1. S = √2 and P = 1/3.
  2. x² − √2 x + 1/3. Here the x is outside the root: the term is √2 times x.
  3. Check: −b/a = √2 and c/a = 1/3.

Answerx² − √2 x + 1/3

(iii) 0, √5

  1. S = 0 and P = √5.
  2. x² − 0·x + √5 = x² + √5, because the x term drops out when S = 0.
  3. Check: −b/a = 0 and c/a = √5.
  4. This polynomial is never 0 for any real x, so it has no real zeroes. The question only asks for the sum and product as −b/a and c/a, and these match.

Answerx² + √5

(iv) 1, 1

  1. S = 1 and P = 1.
  2. x² − 1·x + 1 = x² − x + 1.
  3. Check: −b/a = 1 and c/a = 1.
  4. Like (iii), this polynomial has no real zeroes, but −b/a and c/a are the given numbers.

Answerx² − x + 1

(v) −1/4, 1/4

  1. S = −1/4 and P = 1/4.
  2. x² − (−1/4)x + 1/4 = x² + x/4 + 1/4. Subtracting a negative number is the same as adding.
  3. Check: −b/a = −1/4 and c/a = 1/4.
  4. This one also has no real zeroes, but −b/a and c/a are the given numbers.

Answerx² + x/4 + 1/4

(vi) 4, 1

  1. S = 4 and P = 1.
  2. x² − 4x + 1.
  3. Check: −b/a = 4 and c/a = 1.

Answerx² − 4x + 1

Watch this explained “Run it backwards”, 9:28 into What the sum and product of two zeroes reveal about a, b and c

Every question here was solved twice, separately, by two different AI models, and each answer was put back into the question by a computer program to check it. Where the two disagreed, a stronger model solved it again and the computer check had to pass on its answer. A question about reasoning rather than a number is shown as “one way to think about it”, and anything not yet proven says so instead of guessing. Each answer links to the moment in the video that teaches it.

We quote only enough of each question to find it: keep your NCERT book open, or open this chapter in NCERT’s PDF. Spotted a mistake? Tell us.

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