Class 10 · Chapter 2 · Algebra unit (20 of 80 marks)

Polynomials Class 10: notes and important questions

Q: What is in CBSE Class 10 Maths Chapter 2, Polynomials?

The topics are: Geometrical meaning of the zeroes, Zeroes of a polynomial, Relationship between zeroes and coefficients, Forming a quadratic polynomial. This page has revision notes for each, and 41 board-style practice questions with step-marked solutions.

Q: How many marks is Polynomials worth?

It is in the Algebra unit, which carries 20 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Q: Where do students lose marks in Polynomials?

Sign slip in the sum of zeroes: writing b/a instead of −b/a. Fix: write 'α + β = −b/a = …' with the formula before substituting. Forming the polynomial as x² + Sx + P. Fix: memorise k[x² − (sum)x + (product)] and include 'k' when asked for 'a' quadratic polynomial.

Q: Are the Polynomials important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Q: Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 41 board-style questions.

Practise this chapter Join freeFree account, no card: +5 practice questions on every chapter, with instant feedback.

41 questions
13 multiple choice, 3 assertion–reason, 9 very short answer, 8 short answer, 5 long answer, 3 case study
About 8 hours to master

Algebra unit: 20 of 80 theory marks (Polynomials, Pair of Linear Equations, Quadratic Equations, Arithmetic Progressions).

NCERT textbook: Chapter 2 on ncert.nic.in (PDF)

Revision notes

Polynomials — revision notes

1. Basics

Degree = highest power of the variable. Linear (degree 1), quadratic (degree 2), cubic (degree 3).
A zero of $p (x)$ is a number $k$ with $p (k) = 0$ . A polynomial of degree $n$ has at most $n$ zeroes.

2. Geometrical meaning of zeroes

The zeroes of $p (x)$ are the $x$ -coordinates of the points where the graph of $y = p (x)$ meets the $x$ -axis.
$y = a x + b$ is a straight line: exactly one zero, $- \frac{b}{a}$ .
$y = a x^{2} + b x + c$ is a parabola, opening upward if $a > 0$ and downward if $a < 0$ . It can meet the $x$ -axis at 2 points, touch it at 1 point, or miss it (0 zeroes).

3. Zeroes and coefficients (quadratic $a x^{2} + b x + c$ )

$α + β = - \frac{b}{a}, α β = \frac{c}{a}$

Useful identities: $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ ; $(α - β)^{2} = (α + β)^{2} - 4 α β$ ; $α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$ ; $\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$ .

4. Forming a quadratic polynomial

With sum $S$ and product $P$ of zeroes: $k [x^{2} - S x + P]$ , $k \neq 0$ . Multiply by a suitable $k$ to clear fractions.

Worked example 1

Zeroes of $3 x^{2} + 5 x - 2$ : split $- 6 = 6 \times (- 1)$ : $3 x^{2} + 6 x - x - 2 = (3 x - 1) (x + 2)$ . Zeroes $\frac{1}{3}, - 2$ . Check: sum $- \frac{5}{3} = - \frac{b}{a}$ , product $- \frac{2}{3} = \frac{c}{a}$ .

Worked example 2

Polynomial with zeroes $3 \pm \sqrt{2}$ : $S = 6$ , $P = 9 - 2 = 7$ , so $x^{2} - 6 x + 7$ .

Worked example 3

If $α, β$ are zeroes of $x^{2} - 4 x + 1$ , find $α^{2} + β^{2}$ : $16 - 2 = 14$ .

Common errors

Sign slip: the sum of zeroes is $- b / a$ , not $b / a$ .
Writing $x^{2} + S x + P$ instead of $x^{2} - S x + P$ .
Stating that a quadratic “has exactly 2 zeroes” — it may have 0, 1 or 2 real zeroes.
When verifying the relationship, forgetting to compare with $- b / a$ and $c / a$ explicitly.

Board-exam tips

“Find zeroes and verify” questions are 2–3 marks: factorise by splitting the middle term, then show both checks.
For expressions in $α, β$ , never solve for the zeroes unless asked — use the identities.
Graph-reading MCQs: count intersections with the $x$ -axis only (not the $y$ -axis).

Topics in this chapter: Geometrical meaning of the zeroes · Zeroes of a polynomial · Relationship between zeroes and coefficients · Forming a quadratic polynomial.

Formulas for this chapter: Chapter 2 on the Class 10 formula sheet (free PDF and A3 poster).

Polynomials Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 178 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceRelationship between zeroes and coefficients

$3$ is one zero of $3 x^{2} - 10 x + k$ . The other zero is

Show worked answer

$27 - 30 + k = 0 \Rightarrow k = 3$ ; product of zeroes $= \frac{k}{3} = 1$ , so the other zero is $\frac{1}{3}$ A1
Correct option: (a) $\frac{1}{3}$

Q2·1 mark·Multiple choiceRelationship between zeroes and coefficients

If $α$ and $β$ are the zeroes of $x^{2} - 4 x + 1$ , then $α^{2} + β^{2}$ equals

Q3·1 mark·Assertion–reasonZeroes of a polynomial

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The polynomial $x^{2} - 6 x + 9$ has two equal zeroes.

Reason (R): Every quadratic polynomial has two distinct real zeroes.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerRelationship between zeroes and coefficients

Find the zeroes of $3 x^{2} - 27$ . Check that their sum is $- \frac{b}{a}$ and their product is $\frac{c}{a}$ .

Show worked answer

Answer:

3

and

- 3

$3 x^{2} - 27 = 3 (x - 3) (x + 3)$ , so the zeroes are $3$ and $- 3$ M1
Sum $= 0 = - \frac{0}{3}$ ; product $= - 9 = \frac{- 27}{3}$ A1

Q5·2 marks·Short answerForming a quadratic polynomial

Find a quadratic polynomial with integer coefficients whose zeroes are $- \frac{2}{3}$ and $\frac{1}{2}$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerRelationship between zeroes and coefficients

$α$ and $β$ are the zeroes of $2 x^{2} - 3 x - 6$ . Without finding the zeroes, find $\frac{1}{α} + \frac{1}{β}$ and $α^{2} β + α β^{2}$ .

Show worked answer

Answer:

- \frac{1}{2}

and

- \frac{9}{2}

$α + β = \frac{3}{2}$ and $α β = - 3$ A1
$\begin{aligned} \frac{1}{α} + \frac{1}{β} & = \frac{α + β}{α β} \\ = \frac{3 / 2}{- 3} \\ = - \frac{1}{2} \end{aligned}$ A1
$\begin{aligned} α^{2} β + α β^{2} & = α β (α + β) \\ = - 3 \times \frac{3}{2} \\ = - \frac{9}{2} \end{aligned}$ A1

Q7·3 marks·Short answerRelationship between zeroes and coefficients

Factorise $6 x^{2} - 13 x + 6$ to find its zeroes. Then check that their sum and product agree with the coefficients.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerForming a quadratic polynomial

The zeroes $α$ and $β$ of $x^{2} - 9 x + k$ satisfy $α - β = 3$ . Find $k$ . Hence find a quadratic polynomial whose zeroes are $\frac{α}{β}$ and $\frac{β}{α}$ .

Show worked answer

Answer:

k = 18

;

2 x^{2} - 5 x + 2

$α + β = 9$ and $α β = k$ M1
With $α - β = 3$ : $α = 6$ , $β = 3$ A1
$k = α β = 18$ A1
$\frac{α}{β} = 2$ , $\frac{β}{α} = \frac{1}{2}$ : sum $= \frac{5}{2}$ , product $= 1$ M1
Polynomial: $x^{2} - \frac{5}{2} x + 1$ , or $2 x^{2} - 5 x + 2$ A1

Q9·5 marks·Long answerRelationship between zeroes and coefficients

$α$ and $β$ are the zeroes of $p (x) = 2 x^{2} + 5 x + k$ , and $α^{2} + β^{2} + α β = \frac{21}{4}$ . Find $k$ and the zeroes of $p (x)$ .

Section E style · Case study (4 marks)

Q10·4 marks·Case studyGeometrical meaning of the zeroes

Flower bed. The area of a rectangular flower bed is $p (x) = x^{2} + x - 12$ square metres, where its length and breadth (in metres) are the two linear factors of $p (x)$ .

Factorise $p (x)$ to write the length and the breadth in terms of $x$ . [1 mark]
Find the zeroes of $p (x)$ . [1 mark]
The length of the bed is $9$ m. Find $x$ and the area of the bed. [2 marks]

Show worked answer

$x^{2} + 4 x - 3 x - 12 = (x + 4) (x - 3)$ : length $(x + 4)$ m, breadth $(x - 3)$ m A1
$(x + 4) (x - 3) = 0 \Rightarrow x = - 4$ or $x = 3$ A1
$x + 4 = 9 \Rightarrow x = 5$ ; breadth $= 5 - 3 = 2$ m M1
Area $= 9 \times 2 = 18 m^{2}$ (check: $p (5) = 25 + 5 - 12 = 18$ ) A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can find the zeroes of a quadratic by splitting the middle term and read the number of zeroes from a graph.

Read first: 1. Basics; 2. Geometrical meaning of zeroes 9 practice questions · checkpoint: 4 questions, 8 marks, pass 80%

Practise step 1

Step 2

Board standard

You can use sum = −b/a and product = c/a confidently, verify the relationship, and build a quadratic from given zeroes.

Read first: 3. Zeroes and coefficients (quadratic ax² + bx + c); 4. Forming a quadratic polynomial; Worked examples 1-2 16 practice questions · checkpoint: 4 questions, 9 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write full-marks long answers and case studies, including completing a table, plotting the graph and reading off the zeroes.

Read first: 2. Geometrical meaning of zeroes; 3. Zeroes and coefficients; Board-exam tips 6 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle expressions like α² + β² or α/β + β/α without solving, and build new polynomials whose zeroes are changed versions of old ones.

Read first: 3. Zeroes and coefficients; 4. Forming a quadratic polynomial; Worked example 3 8 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 41 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceZeroes of a polynomial

The zeroes of $x^{2} - 2 x - 15$ are

(a) $3, - 5$
(b) $- 3, - 5$
(c) $3, 5$
(d) $- 3, 5$

Q2·1 mark·Multiple choiceZeroes of a polynomial

If $2$ is a zero of $3 x^{2} - 8 x + k$ , then the value of $k$ is

(a) $4$
(b) $- 4$
(c) $28$
(d) $- 28$

Q3·1 mark·Multiple choiceZeroes of a polynomial

Which of the following is a quadratic polynomial?

(a) $x^{2} + \frac{1}{x}$
(b) $\sqrt{x} + 4$
(c) $3 - 2 x + \sqrt{2} x^{2}$
(d) $x^{3} - x$

Worked solution·2 marks·Very short answerRelationship between zeroes and coefficients

Find the zeroes of $6 x^{2} - 7 x - 3$ and verify the relationship between the zeroes and the coefficients.

Full worked solution Free, no sign-in

$6 x^{2} - 9 x + 2 x - 3 = (2 x - 3) (3 x + 1)$ ; zeroes $\frac{3}{2}, - \frac{1}{3}$ A1
Sum $= \frac{3}{2} - \frac{1}{3} = \frac{7}{6} = - \frac{- 7}{6}$ ; product $= - \frac{1}{2} = \frac{- 3}{6}$ ✓ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I121 · Problem I24 · Problem I106 · Problem I113. All 29 →

Knowledge organiser: Polynomials on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Polynomials

Sign slip in the sum of zeroes: writing b/a instead of −b/a. Fix: write 'α + β = −b/a = …' with the formula before substituting.
Forming the polynomial as x² + Sx + P. Fix: memorise k[x² − (sum)x + (product)] and include 'k' when asked for 'a' quadratic polynomial.
In 'find the zeroes and verify', stopping after the zeroes. Fix: show both checks, sum vs −b/a and product vs c/a, each on its own line; they carry separate marks.
Solving for α and β when the question only needs α² + β² or 1/α + 1/β. Fix: use identities (α² + β² = (α + β)² − 2αβ), which is shorter and is what the scheme rewards.
Graph MCQs: counting where the curve meets the y-axis, or counting a touching point twice. Fix: count only distinct points on the x-axis.
Surd coefficients (e.g. 2√3x² − 5x + √3): splitting the middle term with the wrong product. Fix: multiply a × c first (here 6) and look for factors of that number.

Exam strategy: how answers are step-marked · competency-based questions

Polynomials in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 2, 19, 22
Sample paper 2 (Board standard): questions 1, 2
Sample paper 3 (Above board): questions 2, 3, 22
Sample paper 4 (Above board): questions 2, 3
Sample paper 5 (95+ challenge): questions 2, 22

Want it against the clock? Take a timed 30-mark chapter test on Polynomials, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Polynomials: common questions

What is in CBSE Class 10 Maths Chapter 2, Polynomials?

The topics are: Geometrical meaning of the zeroes, Zeroes of a polynomial, Relationship between zeroes and coefficients, Forming a quadratic polynomial. This page has revision notes for each, and 41 board-style practice questions with step-marked solutions.

How many marks is Polynomials worth?

It is in the Algebra unit, which carries 20 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Polynomials?

Sign slip in the sum of zeroes: writing b/a instead of −b/a. Fix: write 'α + β = −b/a = …' with the formula before substituting. Forming the polynomial as x² + Sx + P. Fix: memorise k[x² − (sum)x + (product)] and include 'k' when asked for 'a' quadratic polynomial.

Are the Polynomials important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Work through the Route to 95 with your progress saved

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Revision notes

Polynomials — revision notes

1. Basics

2. Geometrical meaning of zeroes

3. Zeroes and coefficients (quadratic ax2+bx+c)

4. Forming a quadratic polynomial

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Polynomials Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Polynomials

Polynomials in our sample papers

Polynomials: common questions

Work through the Route to 95 with your progress saved

3. Zeroes and coefficients (quadratic $a x^{2} + b x + c$ )