EduroDownload
Class 10 · NCERT Maths

Chapter 2

Polynomials

Use zeros of polynomials and relationships between coefficients and roots. Learn the ideas, see how they connect, and practise until you can choose the next step without help.

ZerosQuadraticsGraphs

Reviewed by Eduro educators · Built for independent practice

Study board

Quick reference
Zerop(α) = 0
Quadraticax² + bx + c
Sum of zeroesα + β = −b/a
Productαβ = c/a
Cubic productαβγ = −d/a
Graphx-intercepts

NCERT chapter notes

Polynomials, clearly explained

Polynomials connects an algebraic expression with its graph and its zeroes. A number is a zero of p(x) when substituting that number makes p(x) equal to zero; on a graph, these are the x-coordinates where y = p(x) meets the x-axis.

The scoring core of this chapter is the relationship between the zeroes of a polynomial and its coefficients. Students should be able to move in both directions: find sums and products from a given polynomial, and form a polynomial when its zeroes are known.

Core concepts

Key concepts and formulae

01

Polynomial and degree

A polynomial in x has non-negative integer powers of x. The degree is the highest power with a non-zero coefficient: 5x³ − 2x + 7 has degree 3. Expressions such as 1/x or √x are not polynomials in x.

02

Zeroes and the graph

A value k is a zero of p(x) if p(k) = 0. Geometrically, a zero is an x-coordinate where the graph y = p(x) intersects or touches the x-axis. A linear polynomial has one zero; a quadratic can have at most two; a cubic can have at most three.

03

Quadratic relationship

For ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a. Write the polynomial in standard form first, and keep the sign of b when substituting.

04

Cubic relationship

For ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = −b/a; αβ + βγ + γα = c/a; and αβγ = −d/a.

05

Forming a polynomial

If α and β are zeroes, a quadratic polynomial is k[x² − (α + β)x + αβ], where k is any non-zero constant. When a monic polynomial is required, take k = 1.

06

A reliable verification

After finding or constructing zeroes, substitute them into the polynomial. Each should give zero. Also compare their sum and product with the coefficient relationships to catch sign errors.

Exam practice

Important questions, with worked answers

Use each answer to check the method and wording—not as a paragraph to memorise.

Q1Find the zeroes of x² − 7x + 12 and verify the relationship with the coefficients.

Answer

x² − 7x + 12 = (x − 3)(x − 4), so the zeroes are 3 and 4. Their sum is 7 = −(−7)/1, and their product is 12 = 12/1.

Q2Find a quadratic polynomial whose zeroes are 2 and −5.

Answer

The sum is −3 and the product is −10. A monic polynomial is x² − (sum)x + product = x² + 3x − 10. Any non-zero multiple of this polynomial has the same zeroes.

Q3If one zero of 2x² + 5x + k is the reciprocal of the other, find k.

Answer

If the zeroes are reciprocals, their product is 1. For 2x² + 5x + k, product = k/2. Therefore k/2 = 1, so k = 2.

Q4The sum and product of the zeroes of a quadratic polynomial are 4 and −5. Form the polynomial.

Answer

A monic quadratic is x² − (sum)x + product = x² − 4x − 5. It factorises as (x − 5)(x + 1), confirming zeroes 5 and −1.

Q5For 2x³ − 3x² − 11x + 6, state the sum and product of its three zeroes.

Answer

Here a = 2, b = −3, and d = 6. The sum is −b/a = 3/2. The product is −d/a = −3.

Q6Can a quadratic polynomial have three distinct zeroes? Explain geometrically.

Answer

No. The graph of a quadratic polynomial is a parabola, which can intersect the x-axis at at most two points. Therefore a quadratic has at most two distinct real zeroes.

Fast recall

Key terms to remember

Polynomial: An algebraic expression made from variables with non-negative integer powers and numerical coefficients.

Degree: The highest power of the variable with a non-zero coefficient.

Zero: A value k for which p(k) = 0; graphically, an x-axis intersection or touch point.

Coefficient: The numerical factor multiplying a variable term, including its sign.

Monic polynomial: A polynomial whose leading coefficient is 1.

A repeatable method

How to learn Polynomials properly

  1. 1

    Start with the chapter promise

    Use zeros of polynomials and relationships between coefficients and roots. Before solving, the student should be able to say what the chapter is trying to teach and which kind of problem it helps solve.

  2. 2

    Build the core vocabulary

    The important words for this chapter are Zeros, Quadratics and Graphs. Eduro should make the student define each one in simple language, then use it in a question or explanation.

  3. 3

    Move from recognition to recall

    Recognition means the answer makes sense after seeing it. Recall means the student can produce the next step independently. This page is built for recall, because that is what tests reward.

  4. 4

    Close the loop with practice

    A strong study session ends with mixed numericals, not only reading. The student should solve, review the mistake, and then attempt a similar question before moving on.

Before the test

Where students usually lose marks

Look for the first wrong decision—not only the final wrong answer. That is the mistake worth repairing.

Mistake 1

Knowing Zeros only after seeing the solution

This is the most common hidden gap. The student feels confident while reading, but cannot choose the starting step alone. Eduro should ask a short diagnostic question before explaining the method.

Mistake 2

Treating Polynomials as a memory chapter

Even memory-heavy chapters need reasoning. A memorised line becomes fragile when the question changes. The student should explain why the answer works, not only what the answer is.

Mistake 3

Skipping the checking step

calculation slip usually survives because the student finishes the answer and moves on. Eduro should make review part of the answer: what was asked, what was used, and whether the final response fits.

Try it with Eduro

Practice that builds real confidence

Say it naturally, as you would to a patient teacher. Each prompt below is designed to make understanding visible.

Prompt 1

“Ask Eduro to explain Polynomials through Zeros before showing any solved answer.”

Prompt 2

“Create five questions that separately test Zeros, Quadratics and Graphs.”

Prompt 3

“Give one wrong answer from Polynomials and ask the student to find the first incorrect step.”

Prompt 4

“End with a mixed mini-test where Eduro does not reveal which skill is being tested.”

For parents

A better way to ask, “Is the chapter done?”

For Class 10 Maths, a good sign is not that the child says 'Polynomials is done.' A better sign is that they can explain Zeros, solve one fresh question, and correct one mistake without panic.

Created by

Eduro Editorial Team

CBSE learning research and AI tutoring design

Reviewed by

Eduro Academic Review

Curriculum and accuracy review

Freshness

Reviewed for the 2025-26 CBSE/NCERT study cycle

Reviewed quarterly and after major CBSE/NCERT updates

Quick help

Common questions

How can I study Polynomials for NCERT Class 10?

Start with the NCERT examples, understand the key ideas in Zeros, Quadratics, Graphs, then practice exercise questions and ask Eduro where you get stuck.

Can Eduro help with Polynomials?

Yes. Eduro can explain Polynomials step by step, answer follow-up doubts, and help students practice related Maths questions.