Find the zeroes of the polynomial p(x) = 3x² - 2x - 1 and verify the relationship between the zeroes of p(x) and the coefficients of p(x).

MathsCBSE Class 10 · Basic · Board exam 2026, set 430/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2026: question 27

Set in the CBSE Class 10 board exam of 2026, paper 430/2/1, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerPolynomials

Find the zeroes of the polynomial $p (x) = 3 x^{2} - 2 x - 1$ and verify the relationship between the zeroes of $p (x)$ and the coefficients of $p (x)$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/2/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Split the middle term: find two numbers with product 3 × (−1) = −3 and sum −2.

Hint 2

After finding the zeroes, compare their sum with −b/a and product with c/a.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$p (x) = 3 x^{2} - 3 x + x - 1 = 3 x (x - 1) + 1 (x - 1) = (3 x + 1) (x - 1)$ . Zeroes: $x = - \frac{1}{3}$ and $x = 1$ .
1 mark
Sum of zeroes $= - \frac{1}{3} + 1 = \frac{2}{3}$ ; and $\frac{- coefficient of x}{coefficient of x ^{2}} = \frac{- ( - 2 )}{3} = \frac{2}{3}$ . Verified.
1 mark
Product of zeroes $= - \frac{1}{3} \times 1 = - \frac{1}{3}$ ; and $\frac{constant term}{coefficient of x ^{2}} = \frac{- 1}{3}$ . Verified.
1 mark

Answer: Zeroes are −1/3 and 1; sum = 2/3 = −b/a, product = −1/3 = c/a

Where the marks are

Factorises and finds zeroes −1/3, 1 (1 mark)
Verifies sum of zeroes = −b/a (1 mark)
Verifies product of zeroes = c/a (1 mark)

Revise the chapter

Chapter: Polynomials (notes, formulas, where marks are lost and every past question)
Topic: Relationship Between Zeros and Coefficients

Practise this chapterBoard-style questions from Polynomials, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 430/2/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2026: question 27

Hints

Step-by-step solution

Revise the chapter

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