Polynomials — Class 10 Notes

CBSE 2027 — dates, rules and official papers →

Maths · Ch 2Class 10 · CBSE 2026–27Medium weightage

Polynomials

What the board actually asks: finding the zeroes of a quadratic polynomial and verifying the sum/product relationship with its coefficients, reading the number of zeroes off a graph (the geometrical meaning), and forming a quadratic from a given sum and product of zeroes.

NCERT VERBATIM sourced, word-for-wordAUTHORED LazyTopper teaching scaffolding

The Big Idea AUTHORED

Zeroes are where the graph lands on the x-axis.

A polynomial's zeroes are the values of $x$ that make it $0$ - and geometrically, they are exactly the points where the graph $y = p (x)$ crosses the $x$ -axis. This short chapter ties three views of one idea together: the algebra (factorise to find the zeroes), the geometry (count the $x$ -axis crossings), and the coefficients (for a quadratic, $α + β = - \frac{b}{a}$ and $α β = \frac{c}{a}$ ). Master the quadratic case - that is what the board assesses.

Formulae

Quadratic polynomial

a x^{2} + b x + c, a \neq = 0

Tip: Degree 2; if a = 0 it is not quadratic.

Zero

p (k) = 0

Tip: A zero is the x-coordinate of the point where the graph meets the x-axis.

Sum of zeroes

α + β = - \frac{b}{a}

Tip: α+β = −b/a: the sign flips against the x-coefficient.

Product of zeroes

α β = \frac{c}{a}

Tip: αβ = c/a.

Build from zeroes

x^{2} - (α + β) x + α β

Tip: Write "x² − (α+β)x + αβ" and say any non-zero multiple also works.

Zeroes of a Quadratic from its Graph

The graph of a quadratic polynomial is a parabola. If it cuts the x-axis twice there are 2 zeroes; if it just touches the x-axis there is 1 (repeated) zero; if it never meets the x-axis there is no real zero. The parabola opens upwards if

a > 0

and downwards if

a < 0

. A quadratic polynomial has at most 2 zeroes.

Expressions in

α

and

β

α^{2} + β^{2} = (α + β)^{2} - 2 α β

\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}

\frac{α}{β} + \frac{β}{α} = \frac{α ^{2} + β ^{2}}{α β}

Verifying the Relationship

Find the zeroes, then check that their sum equals

- \frac{b}{a}

and their product equals

\frac{c}{a}

Shortcuts & memory tricks →

Definitions the board rewards NCERT VERBATIM

State these in NCERT’s exact words

Degree of a polynomial

NCERT verbatim — memorise this exact wording· Ch 2,

“Recall that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x).”

What it means: the highest power of the variable.

4 x + 2

has degree 1;

2 y^{2} - 3 y + 4

has degree 2. Degree tells you the polynomial's family - and caps how many zeroes it can have.

Quadratic polynomial

NCERT verbatim — memorise this exact wording· Ch 2,

“More generally, any quadratic polynomial in x is of the form ax2 + bx + c, where a, b, c are real numbers and a ≠ 0.”

What it means: a degree-2 polynomial,

a x^{2} + b x + c

with

a \neq = 0

. This is the case the whole chapter (and the board) focuses on.

Zero of a polynomial

NCERT verbatim — memorise this exact wording· Ch 2,

“More generally, a real number k is said to be a zero of a polynomial p(x), if p(k) = 0.”

What it means: a value of

x

that makes the polynomial equal

0

. Same idea as a root of the equation

p (x) = 0

. Note: a zero is not the value

p (0)

Geometrical meaning of the zeroes

NCERT verbatim — memorise this exact wording· Ch 2,

“the zeroes of a quadratic polynomial ax2 + bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax2 + bx + c intersects the x-axis.”

What it means: draw

y = p (x)

and look at where it meets the

x

-axis - each crossing is a zero. This is why the number of zeroes can be read straight off a graph.

Key terms — NCERT verbatim (state word-for-word)

Value p(k): If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k). (Ch 2, )
Linear polynomial: A polynomial of degree 1 is called a linear polynomial. (Ch 2, )
Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial. (Ch 2, )

Must-know concepts AUTHORED FRAMING

…and where they’re tested

Zeroes of a polynomial

A zero of

p (x)

is any value

k

with

p (k) = 0

- substitute it and the polynomial collapses to

0

. For a quadratic, factorise

a x^{2} + b x + c

into two linear factors and set each to zero; the two values you get are the zeroes.

NCERT Fig 2.2Graph of y = x^2 - 3x - 4 · the curve cuts the x-axis at x = -1 and x = 4 - its two zeroes.

1-mark 'is

k

a zero of

p (x)

?' and 2-mark 'find the zeroes'.

Geometrical meaning - zeroes are x-axis crossings

The zeroes of

p (x)

are exactly the

x

-coordinates where the graph

y = p (x)

crosses the

x

-axis. A parabola

y = a x^{2} + b x + c

can meet the axis at two points (two distinct zeroes), touch it at one point (two equal zeroes), or miss it entirely (no zero).

Figurey = p(x) - the zeroes are where the parabola meets the x-axis

two zeroes

cuts at x = -1 and x = 4 (NCERT's y = x^2 - 3x - 4)

one point

touches at x = 2 (two equal zeroes)

no zero

stays above the x-axis

Left: two distinct zeroes (cuts twice) · Middle: two equal zeroes (just touches) · Right: no real zero (never meets).

1-mark MCQ 'number of zeroes from the graph' (Exercise 2.1 style).

A quadratic has at most two zeroes

Degree bounds the count: the graph of a degree-

n

polynomial meets the

x

-axis at most

n

times, so a quadratic has at most two zeroes and a linear polynomial has exactly one.

Reasoning / MCQ on the maximum number of zeroes.

Zeroes and coefficients (quadratic)

For zeroes

α, β

a x^{2} + b x + c

α + β = - \frac{b}{a}

(sum) and

α β = \frac{c}{a}

(product). This lets you verify computed zeroes without solving again.

2-3 mark 'find the zeroes and verify the relationship' - the chapter's staple.

Building a quadratic from its zeroes

Given the sum

s

and product

p

of the zeroes, a quadratic is

x^{2} - s x + p

(and any real multiple

k

of it also works). So sum

- 3

, product

2

gives

x^{2} + 3 x + 2

2-3 mark 'form the quadratic polynomial'.

Worked examples — real solved problems NCERT VERBATIM

Solved exactly as the source does them

Every problem is a real sourced item, quoted verbatim with its page; the step-by-step decoding and mark-logic are LazyTopper-authored coaching.

Ch 2, Example 2,

“Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.”

Factorise.

x^{2} + 7 x + 10 = (x + 2) (x + 5)

.Concept: factorise into two linear factors

1 mark

Set each factor to zero.

x^{2} + 7 x + 10 = 0

when

x + 2 = 0

x + 5 = 0

, i.e.

x = - 2

x = - 5

. So the zeroes are

- 2

and

- 5

.Concept: read off both zeroes

1 mark

Verify sum and product.Sum

= - 2 + (- 5) = - 7 = - \frac{7}{1} = - \frac{b}{a}

; product

= (- 2) (- 5) = 10 = \frac{10}{1} = \frac{c}{a}

.Concept: check against the coefficients

1 mark

Mark-scheme logic (authored): 3-mark shape: 1 for the factorisation, 1 for stating BOTH zeroes, 1 for verifying sum

= - b / a

and product

= c / a

. The verification is explicitly asked - skip it and the last mark is gone.

Ch 2, Example 3,

“Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the coefficients.”

Ch 2, Example 4,

“Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.”

Ch 2,

“Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x2 – 2x – 8”

Problem-solving strategies AUTHORED

'Find the zeroes'?Factorise the quadratic into two linear factors, set each to zero, and read off BOTH zeroes.

'...and verify the relationship'?Check

α + β = - \frac{b}{a}

and

α β = \frac{c}{a}

- it is a separate mark, never skip it.

'Form a quadratic with these zeroes'?Use

x^{2} - (sum) x + (product)

; add that any real

k

-multiple also fits.

Graph given, 'how many zeroes'?Count the points where the curve crosses the

x

-axis - that is the number of zeroes.

'Is

k

a zero of

p (x)

?'Compute

p (k)

; it is a zero exactly when

p (k) = 0

Common mistakes that cost marks

Where students lose easy marks

Giving only one zero of a quadratic - always solve both linear factors and state both values.

Sign slip in the sum:

α + β = - \frac{b}{a}

carries a minus; the product

α β = \frac{c}{a}

does not.

Confusing a 'zero' with

p (0)

- a zero is where

p (x) = 0

, not where

x = 0

Counting a touch-point as two crossings - when the parabola just touches the axis, that is two equal zeroes (one point), not two distinct ones.

Formulae

Quadratic polynomial

a x^{2} + b x + c, a \neq = 0

Tip: Degree 2; if a = 0 it is not quadratic.

Zero

p (k) = 0

Tip: A zero is the x-coordinate of the point where the graph meets the x-axis.

Sum of zeroes

α + β = - \frac{b}{a}

Tip: α+β = −b/a: the sign flips against the x-coefficient.

Product of zeroes

α β = \frac{c}{a}

Tip: αβ = c/a.

Build from zeroes

x^{2} - (α + β) x + α β

Tip: Write "x² − (α+β)x + αβ" and say any non-zero multiple also works.

Zeroes of a Quadratic from its Graph

a > 0

and downwards if

a < 0

. A quadratic polynomial has at most 2 zeroes.

Expressions in

α

and

β

α^{2} + β^{2} = (α + β)^{2} - 2 α β

\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}

\frac{α}{β} + \frac{β}{α} = \frac{α ^{2} + β ^{2}}{α β}

Verifying the Relationship

Find the zeroes, then check that their sum equals

- \frac{b}{a}

and their product equals

\frac{c}{a}

Shortcuts & Tricks

If the zeroes are negatives of each other (α+β = 0) the x-term is missing (b = 0).

MCQ / check only: don't write this as working

Zeroes reciprocal of each other ⇔ αβ = 1 ⇔ c = a.

MCQ / check only: don't write this as working

1/α + 1/β = −b/c (c ≠ 0).

MCQ / check only: don't write this as working

Show the line α²+β² = (α+β)² − 2αβ, then substitute.

Safe in written answers

Count x-axis crossings; a touch means equal zeroes.

MCQ / check only: don't write this as working

Memory Tricks

Jod = minus Bhai upon Aloo. Gunaa = Chacha upon Aloo.

Sum = −b/a, product = c/a. Jod = sum, Gunaa = product; Bhai = b, Aloo = a, Chacha = c; minus applies to the sum only.

Do baar kate = do zero · ek baar chhue = ek zero (do barabar) · kabhi na mile = real zero nahi.

Parabola cases. 2 crossings = 2 zeroes; touch = 1 repeated zero; no meet = none.

x² minus (Jod)x plus Gunaa.

x² − (α+β)x + αβ. minus goes with the sum, plus with the product.

Mindmap

- - p(k) = 0 makes k a zero
  - Factorise, then set each factor to 0
  - Linear: one zero; quadratic: at most two
- - Zeroes = x-axis crossings of y = p(x)
  - Two points to two distinct zeroes
  - Touches to two equal zeroes
  - Misses to no zero
- - Sum = -b/a
  - Product = c/a
  - Verify computed zeroes
- - $x^{2} - (sum) x + product$
  - Any real k-multiple also works
- - Give BOTH zeroes
  - Minus sign in the sum
  - A 'zero' is not p(0)

Zeroes & coefficients of a quadratic - why sum = -b/a and product = c/a

The relationship you use to verify zeroes is not a rule to memorise blindly - NCERT derives it by writing the quadratic in terms of its factors and comparing coefficients. The 2026-27 syllabus keeps this QUADRATIC relationship (the cubic version is out of scope).

α

and

β

are the zeroes of

p (x) = a x^{2} + b x + c

, then

(x - α)

and

(x - β)

are its factors.

p (x) = a x^{2} + b x + c, a \neq = 0

p (x)

is a constant

k

times the product of those factors.

a x^{2} + b x + c = k (x - α) (x - β)

Expand the right-hand side.

= k x^{2} - k (α + β) x + k α β

Compare the coefficients of

x^{2}

, of

x

, and the constant term on both sides.

a = k, b = - k (α + β), c = k α β

Divide through by

k = a

to isolate the sum and product.

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

b/a and c/a fall straight out - this is exactly the relationship used to verify a pair of zeroes.

3 competency questions from the CBSE board pattern, with the full step-marked solution for each — the marks are shown per step, as the official scheme awards them.

1 markSection A

Which of the following could be the graph of the polynomial (x − 1)^2(x + 2)?

Option graphs (a) and (b): two polynomial curves drawn on x-y axes, each meeting the x-axis

Option graphs (c) and (d): two polynomial curves drawn on x-y axes, each meeting the x-axis

(A) Graph (a)
(B) Graph (b)
(C) Graph (c)
(D) Graph (d)

Step-marked solution

[1 mark] Zeroes of (x − 1)²(x + 2) are x = 1 (repeated, so the graph touches the x-axis there without crossing) and x = −2 (crosses the x-axis); p(0) = (1)(2) = 2. Graph (a) crosses at both −2 and 1, graphs (b) and (d) touch the x-axis at a negative x and do not pass through (0, 2). Graph (c) crosses at −2, touches at 1 and passes through (0, 2), so it is the graph of the polynomial.

5 marksSection D

One condition on the zeroes α and β of p(x) = x² − 6x + k is that the sum of their squares (α² + β²) equals 20. Determine the value of k, then obtain the two zeroes.

Step-marked solution

[1 mark] For x² − 6x + k: sum of zeroes α + β = −(−6)/1 = 6 and product αβ = k/1 = k.
[1 mark] Use the identity α² + β² = (α + β)² − 2αβ = 36 − 2k.
[1 mark] Apply the given condition: 36 − 2k = 20.
[1 mark] Solve: 2k = 16 ⇒ k = 8.
[1 mark] Then p(x) = x² − 6x + 8 = (x − 2)(x − 4), zeroes 2 and 4; check 2² + 4² = 4 + 16 = 20. ✓

5 marksSection D

For p(x) = x² − 8x + k with zeroes α and β, the sum of the reciprocals of the zeroes, 1/α + 1/β, comes to 2/3. Work out k and list both zeroes.

Step-marked solution

[1 mark] For x² − 8x + k: α + β = 8 and αβ = k.
[1 mark] Express the condition: 1/α + 1/β = (α + β)/(αβ) = 8/k.
[1 mark] So 8/k = 2/3.
[1 mark] Cross-multiply: 2k = 24 ⇒ k = 12.
[1 mark] Then p(x) = x² − 8x + 12 = (x − 2)(x − 6), zeroes 2 and 6; check 1/2 + 1/6 = 2/3. ✓

Practise Polynomials competency questions

Read NCERT Chapter 2 on LazyTopper (jemh102)

Official NCERT chapter PDF (jemh102) — ncert.nic.in

Sources: NCERT 2026-27 (tap to view)

Source ledger PROVENANCE

Source edition: NCERT Reprint 2026-27, Class 10 Mathematics, Ch 2 (jemh102.pdf, printed pages 10-23). One row per sourced item — every verbatim, example and NCERT figure traces to its page.

Item	Type	Source
Degree of a polynomial	verbatim	NCERT 2026-27 Ch2
Quadratic polynomial	verbatim	NCERT 2026-27 Ch2
Zero of a polynomial	verbatim	NCERT 2026-27 Ch2
Geometrical meaning of the zeroes	verbatim	NCERT 2026-27 Ch2
Value p(k)	verbatim	NCERT 2026-27 Ch2
Linear polynomial	verbatim	NCERT 2026-27 Ch2
Cubic polynomial	verbatim	NCERT 2026-27 Ch2
Find zeroes + verify · NCERT Example 2	example	NCERT 2026-27 Ch2 Example 2
Irrational zeroes · NCERT Example 3	example	NCERT 2026-27 Ch2 Example 3
Form a quadratic · NCERT Example 4	example	NCERT 2026-27 Ch2 Example 4
Find zeroes + verify · NCERT Exercise 2.2 Q1(i)	example	NCERT 2026-27 Ch2 Exercise 2.2 Q1(i)
NCERT Fig 2.2 (graph of y = x^2 - 3x - 4)	ncert-figure	NCERT 2026-27 Ch2 Fig 2.2 (jemh102 pdf idx 3)

Polynomials · spec-rendered note · NCERT Reprint 2026-27, Class 10 Mathematics, Ch 2 (jemh102.pdf, printed pages 10-23)

Practise writing answers the way examiners mark them → Guide