Chapter 2 of 14

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Rajasthan (RBSE)Class 10 Mathematics★ 3–4 marks · RBSE Class 10 Maths board (Unit — Algebra)

Polynomials

RBSE Class 10 Maths Chapter 2 — degree and types of polynomials, the geometric meaning of a zero (where the graph cuts the x-axis), and the all-important relationship between the zeroes and coefficients of a quadratic (sum = −b/a, product = c/a). Includes forming a quadratic from its zeroes, with fully solved board-pattern problems.

⏱ 34 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Classify polynomials by degree (linear, quadratic, cubic) and identify their general form
2Find the value and the zeroes of a polynomial by substitution and factorisation
3Interpret the zeroes of a polynomial geometrically as where its graph meets the x-axis
4Apply the relationships sum = −b/a and product = c/a for a quadratic's zeroes
5Construct a quadratic polynomial given the sum and product of its zeroes

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Why this chapter matters

The zero–coefficient relationship (sum = −b/a, product = c/a) is one of the most reused tools in Class 10 and Class 11 algebra. In the RBSE board this chapter reliably gives a 'verify/find zeroes' question and a 'form the quadratic from its zeroes' question — quick, near-certain marks.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Polynomials (Class 9)

Factorisation, the factor theorem and remainder theorem.

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Real Numbers (Class 10)

Working comfortably with rational coefficients.

Polynomials — RBSE Class 10 (Mathematics)

A polynomial is just a tidy sum of powers of x — like $2 x^{2} - 3 x + 1$ . Its zeroes are the x-values that make it vanish, and on a graph those are exactly the points where the curve crosses the x-axis. The beautiful surprise of this chapter: the zeroes are secretly encoded in the coefficients, and you can read one off the other without ever solving the equation.

1. Polynomials, degree and types

A polynomial in x is an expression $a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$ where the powers are whole numbers. The highest power is its degree.

Degree	Name	General form
1	Linear	$a x + b$
2	Quadratic	$a x^{2} + b x + c$
3	Cubic	$a x^{3} + b x^{2} + c x + d$

(In each, $a \neq = 0$ .) The word quadratic comes from quadratum, Latin for square — because of the $x^{2}$ term.

2. The value and the zero of a polynomial

The value of a polynomial $p (x)$ at $x = k$ is the number $p (k)$ you get by substituting.

A zero of $p (x)$ is a value $k$ for which $p (k) = 0$ .

Example: for $p (x) = x^{2} - 3 x - 4$ , $p (4) = 16 - 12 - 4 = 0, p (- 1) = 1 + 3 - 4 = 0.$ So 4 and −1 are the zeroes.

A polynomial of degree n has at most n zeroes. A linear polynomial has exactly one, a quadratic at most two, a cubic at most three.

3. Geometric meaning — zeroes on the graph

Plot $y = p (x)$ . The zeroes are precisely the x-coordinates of the points where the graph meets the x-axis.

A linear graph is a straight line — it crosses the x-axis once → 1 zero.
A quadratic graph is a parabola (∪ if $a > 0$ , ∩ if $a < 0$ ). It can cut the x-axis at two points (2 zeroes), touch it at one point (1 repeated zero), or miss it entirely (0 real zeroes).
A cubic can cross up to three times.

This is why "number of zeroes = number of times the graph meets the x-axis" — a favourite RBSE question is to count the zeroes from a given graph.

4. Relationship between zeroes and coefficients

This is the heart of the chapter. For a quadratic $a x^{2} + b x + c$ with zeroes $α$ and $β$ :

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

Sum of zeroes = −(coefficient of x)/(coefficient of x²). Product of zeroes = (constant term)/(coefficient of x²).

Check with $x^{2} - 3 x - 4$ (a = 1, b = −3, c = −4), zeroes 4 and −1:

sum = 4 + (−1) = 3 and −b/a = 3 ✓
product = 4 × (−1) = −4 and c/a = −4 ✓

For a cubic $a x^{3} + b x^{2} + c x + d$ with zeroes $α, β, γ$ : $α + β + γ = - \frac{b}{a}, α β + β γ + γ α = \frac{c}{a}, α β γ = - \frac{d}{a} .$

5. Forming a quadratic from its zeroes

Reverse the relationship. If you know the sum S and product P of the zeroes, the quadratic is:

$x^{2} - (S) x + P = k [x^{2} - (sum) x + (product)]$

Example — a quadratic whose zeroes have sum $- 3$ and product $2$ : $x^{2} - (- 3) x + 2 = x^{2} + 3 x + 2.$ (Check: factorises as $(x + 1) (x + 2)$ , zeroes $- 1, - 2$ , sum $- 3$ , product $2$ . ✓)

This "build the polynomial backwards" step is a standard 2–3 mark board question.

6. Closing thought

The chapter delivers one elegant idea: the coefficients and the zeroes hold the same information, just packaged differently. With $α + β = - b / a$ and $α β = c / a$ you can:

verify zeroes without re-solving,
find one zero when the other is known,
compute things like $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ instantly, and
construct a quadratic to order from its zeroes.

These relationships return immediately in the next chapter (Quadratic Equations) and again in Class 11. For the RBSE board, a sum/product question and a "form the polynomial" question are near-certain — make these formulas reflexive.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Quadratic — sum of zeroes

α + β = −b/a

For ax² + bx + c; −(coeff of x)/(coeff of x²).

Quadratic — product of zeroes

αβ = c/a

(constant)/(coeff of x²).

Form a quadratic

x² − (sum)x + (product)

Reconstruct the polynomial from its zeroes.

Useful identity

α² + β² = (α + β)² − 2αβ

Compute symmetric expressions without finding the zeroes.

Cubic relationships

α+β+γ = −b/a; αβ+βγ+γα = c/a; αβγ = −d/a

For ax³ + bx² + cx + d.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting the negative sign in sum of zeroes = −b/a

✓ Sum is −b/a, NOT b/a. The product is +c/a. Mixing the signs is the most common slip — write both formulas down before substituting.

WATCH OUT

✗ Reading the wrong number of zeroes from a graph

✓ Count how many times the curve actually MEETS the x-axis. A parabola that only touches the axis has one (repeated) zero, not two.

WATCH OUT

✗ Dropping the leading coefficient 'a' when a ≠ 1

✓ In 2x² − 3x + 1, a = 2. Use sum = −(−3)/2 = 3/2 and product = 1/2 — not just −b and c.

WATCH OUT

✗ Writing the formed quadratic as x² − (sum) − (product)

✓ It is x² − (sum)x + (product). The product is added as the constant term, with a sign of plus.

WATCH OUT

✗ Confusing 'value' p(k) with 'zero'

✓ p(k) is the number you get on substituting x = k. k is a ZERO only when that value equals 0.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 2.1

Exercise 2.1

Reading the number of zeroes from given graphs

Questions

Ex 2.2

Exercise 2.2

Finding zeroes and verifying the zero–coefficient relationship; forming quadratics

Questions

Examples (in-text)

Worked zeroes, relationships and graph interpretations

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Degree

State the degree of the polynomial 5x³ − 4x² + 7x − 2.

▶ Show solution

The highest power of x is 3.

✦ Answer: Degree 3 (a cubic polynomial).

Q2EASY· Zero check

Is x = 2 a zero of p(x) = x² − 5x + 6?

▶ Show solution

Step 1 — p(2) = 4 − 10 + 6 = 0.

✦ Answer: Yes, since p(2) = 0, x = 2 is a zero.

Q3EASY· Relationships

For x² − 7x + 10, write the sum and product of its zeroes without solving.

▶ Show solution

Step 1 — a = 1, b = −7, c = 10.

Step 2 — Sum = −b/a = 7; Product = c/a = 10.

✦ Answer: sum = 7, product = 10.

Q4MEDIUM· Find zeroes

Find the zeroes of x² − 2x − 8 and verify the relationship with its coefficients.

▶ Show solution

Step 1 — Factorise: x² − 2x − 8 = (x − 4)(x + 2). Zeroes: 4 and −2.

Step 2 — Sum = 4 + (−2) = 2; −b/a = −(−2)/1 = 2. ✓

Step 3 — Product = 4 × (−2) = −8; c/a = −8/1 = −8. ✓

✦ Answer: zeroes 4 and −2; relationships verified.

Q5MEDIUM· Form quadratic

Find a quadratic polynomial whose zeroes have sum 4 and product 1.

▶ Show solution

Step 1 — Quadratic = x² − (sum)x + (product).

Step 2 — = x² − 4x + 1.

✦ Answer: x² − 4x + 1.

Q6MEDIUM· Leading coefficient

Find the sum and product of the zeroes of 2x² − 5x + 3.

▶ Show solution

Step 1 — a = 2, b = −5, c = 3.

Step 2 — Sum = −b/a = 5/2.

Step 3 — Product = c/a = 3/2.

✦ Answer: sum = 5/2, product = 3/2.

Q7HARD· Symmetric expression

If α and β are the zeroes of x² − 6x + 8, find α² + β².

▶ Show solution

Step 1 — Sum α + β = 6; product αβ = 8.

Step 2 — Use α² + β² = (α + β)² − 2αβ.

Step 3 — = 6² − 2(8) = 36 − 16 = 20.

✦ Answer: α² + β² = 20.

Q8HARD· Find polynomial

Find a quadratic polynomial whose zeroes are 3 + √2 and 3 − √2.

▶ Show solution

Step 1 — Sum = (3 + √2) + (3 − √2) = 6.

Step 2 — Product = (3 + √2)(3 − √2) = 9 − 2 = 7.

Step 3 — Quadratic = x² − (sum)x + product = x² − 6x + 7.

✦ Answer: x² − 6x + 7.

Q9HARD· Unknown coefficient

If one zero of x² − 4x + k is 3, find k and the other zero.

▶ Show solution

Step 1 — Since 3 is a zero, p(3) = 0: 9 − 12 + k = 0 ⇒ k = 3.

Step 2 — Sum of zeroes = −b/a = 4, so other zero = 4 − 3 = 1.

Step 3 — Check: product = c/a = k = 3 = 3 × 1. ✓

✦ Answer: k = 3, other zero = 1.

Q10HARD· Reciprocal zeroes

If α and β are the zeroes of x² − 5x + 6, find a quadratic polynomial whose zeroes are 1/α and 1/β.

▶ Show solution

Step 1 — For x² − 5x + 6: α + β = 5, αβ = 6.

Step 2 — New sum = 1/α + 1/β = (α + β)/(αβ) = 5/6.

Step 3 — New product = 1/(αβ) = 1/6.

Step 4 — Polynomial = x² − (5/6)x + 1/6, or multiplying by 6: 6x² − 5x + 1.

✦ Answer: 6x² − 5x + 1.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Degree gives the type: 1 linear, 2 quadratic, 3 cubic. A degree-n polynomial has at most n zeroes.
•A zero k makes p(k) = 0; geometrically it is where the graph meets the x-axis.
•Quadratic graph = parabola; it can meet the x-axis at 2, 1 or 0 points.
•Quadratic: sum of zeroes = −b/a, product = c/a.
•Form a quadratic: x² − (sum)x + (product), up to a constant multiple.
•α² + β² = (α + β)² − 2αβ; (α − β)² = (α + β)² − 4αβ.
•Cubic: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a.
•Use the leading coefficient a — do not assume a = 1.

Rajasthan (RBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 3–4 marks

Question type	Marks each	Typical count	What it tests
MCQ / very short	1	1	Degree, number of zeroes from a graph, sum/product recall
Short answer	2	1	Find zeroes & verify; form a quadratic from sum/product
Short answer	3	0–1	Symmetric expressions; unknown coefficient; reciprocal zeroes

Prep strategy

Make sum = −b/a and product = c/a reflexive, with the signs correct
Practise reading the number of zeroes off parabola/cubic graphs
Memorise α² + β² = (α+β)² − 2αβ for symmetric-expression questions
Drill 'form the quadratic' both ways: from numeric zeroes and from sum/product

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Projectile paths

The height of a thrown ball follows a quadratic in time; its zeroes are the moments it is at ground level.

Profit and break-even

A quadratic profit model's zeroes are the break-even points where profit is zero — used in business maths.

Designing arches and dishes

Parabolic shapes (bridges, satellite dishes, headlight reflectors) are modelled by quadratic polynomials.

Computer graphics

Curves in fonts and animations are built from polynomial (Bézier) pieces fitted through control points.

Optimisation

The vertex of a quadratic gives the maximum or minimum value — used to optimise area, cost and revenue.

Signal and data fitting

Engineers fit polynomials to data points to model and predict trends smoothly.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Always write sum = −b/a and product = c/a before substituting — guards against sign errors.
When asked to 'verify', show both the factorised zeroes AND the −b/a, c/a check.
For symmetric expressions, convert to (α+β) and αβ before plugging numbers.
Clear fractions in a 'form the polynomial' answer by multiplying through (e.g. 6x² − 5x + 1).
Read graphs carefully — touching the axis is one repeated zero, not two.
State the final polynomial explicitly; an expression left mid-working may lose the last mark.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Vieta's formulas in full generality — relating all elementary symmetric functions of the roots to the coefficients.
Using the discriminant to predict the number and nature of zeroes before solving.
Polynomial division, the factor and remainder theorems for higher-degree polynomials.
Newton's identities linking power sums of roots (α^n + β^n) to the coefficients.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

RBSE Class 10 Board (BSER Ajmer)High — a zeroes/relationship question almost every year

NTSE / state scholarshipMedium — algebra MCQs on zeroes and coefficients

JEE FoundationHigh — Vieta's relations are used throughout Class 11–12 algebra

Maths Olympiad (NMTC/IMO)Medium — symmetric functions of roots are a common theme

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Yes. RBSE prescribes the NCERT Mathematics textbook for Class 10, so the chapter and exercises are identical. RBSE (BSER Ajmer) sets the exam pattern and marking.

If you expand a(x − α)(x − β) = ax² − a(α + β)x + aαβ and compare with ax² + bx + c, the coefficient of x gives b = −a(α + β), so α + β = −b/a. The minus sign comes straight from the expansion.

Count the number of distinct points where the curve crosses (or touches) the x-axis. Each such point is one real zero; a point where it only touches counts as a repeated zero.

Yes. If its parabola does not meet the x-axis at all (it stays entirely above or below), it has no real zeroes — the zeroes are complex, which you study in Class 11.

Yes, that is exactly why we divide by a: sum = −b/a and product = c/a. For 2x² − 5x + 3, the sum is 5/2 and the product is 3/2.

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