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Class 10 Maths Notes

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Polynomials Class 10 Maths Notes

Polynomials is the opening section of Unit II, Algebra, and it deals with only two things: what a zero of a polynomial is, and what the two zeros of a quadratic tell you about its coefficients. The degree tells you how many zeros to expect, the graph turns a zero into a point on the x-axis, and the pair α and β pins down both the sum and the product.

Class:10Subject:MathematicsUnit:IICovers:CBSE 2024-25
6 Key Formulas
DWritten byDeep Narayan
Updated
Key Concept Summary

What is the relationship between the zeros and the coefficients of a quadratic polynomial?

If the zeros of p(x) = ax² + bx + c are α and β, then α + β = −b/a and αβ = c/a. So the sum of the zeros carries the coefficient of x, with its sign changed and divided by the leading coefficient, and the product of the zeros is the constant term divided by the leading coefficient. Both relations have the minus sign on the sum and none on the product.

01

What a Polynomial Is

A polynomial in one variable is a sum of terms in which the only operations are addition of terms and multiplication by whole numbers. No letter may appear in a denominator, and no negative exponent is allowed, so x² + 3x + 5 is a polynomial but 1/x and x⁻¹ are not. Every polynomial in this chapter also has a degree, which is the one piece of information you use most often.

  • 5 is a polynomial of degree 0, since a non-zero constant is a polynomial with no variable part.
  • 3x − 7 is a polynomial of degree 1, and it is therefore called a linear polynomial.
  • x² + 3x + 7 is a polynomial of degree 2, and it is therefore called a quadratic polynomial.
  • x³ − 5x + 11 is a polynomial of degree 3. Note the missing x² term: a missing term is a term with coefficient 0, which does not raise the degree.
  • 1/x and 1 + 1/x are not polynomials, because x appears in a denominator.

Do not use the division algorithm

An older version of this unit carried the division algorithm for polynomials, in which p(x) is split into q(x) times a quotient plus a remainder. That has been removed from the 2024-25 course. Everything the board asks in this chapter is done with the definition of a zero, with factorisation, and with the two relations between zeros and coefficients.
02

Zeros of a Polynomial

Definition

Zero of a polynomial

A real number k is a zero of the polynomial p(x) if p(k) = 0. The value k itself is called the zero; the point (k, 0) is the zero point. So finding the zeros of a polynomial is exactly the work of putting p(x) equal to 0 and solving.

The definition of a zero, and the reason every zeros question begins p(x) = 0.

A polynomial of degree n has at most n zeros. That single fact settles three common questions at a stroke: a linear polynomial has at most one zero, a quadratic has at most two, and if you are told that a cubic has three distinct zeros then it must have exactly three and all its degree is used up. A quadratic whose discriminant-style working gives a repeated value has one zero repeated twice, not two zeros.

  • For p(x) = x² − 5x + 6, put p(x) = 0: x² − 5x + 6 = 0, which is (x − 2)(x − 3) = 0. The zeros are 2 and 3.
  • For p(x) = x² + 4x + 4, put p(x) = 0: (x + 2)² = 0. There is one zero, x = −2, repeated twice.
  • For p(x) = 2x − 9, put p(x) = 0: 2x − 9 = 0, so x = 9/2. A linear polynomial has exactly one zero.
03

The Geometric Meaning of a Zero

Draw the graph of y = p(x). The zeros are the x-coordinates of the points where that graph meets, or touches, the x-axis. Because the graph meets the x-axis only where y = 0, and y = p(x) at that point, the meeting points are exactly (k, 0) for the zeros k. This is why a zero is never called a y-value: it is an x-coordinate.

A zero of p is an x-coordinate of an intersection of the graph with the x-axis.
  • p(x) = x² − 5x + 6 has zeros 2 and 3, so the graph of y = x² − 5x + 6 cuts the x-axis at (2, 0) and (3, 0).
  • Since the leading coefficient is positive the parabola opens upwards, so between the two zeros the graph lies below the x-axis, where p(x) is negative.
  • p(x) = x² + 4x + 4 has the single zero −2 repeated twice, so the graph touches the x-axis at (−2, 0) and turns back. It never crosses.
  • Outside the two zeros the graph lies above the x-axis, where p(x) is positive. This sign picture is worth one mark in any graph-based question.

Cutting or touching, and how to tell

Two distinct zeros means the graph cuts the x-axis twice. One repeated zero means the graph touches the x-axis at a single point and turns. If you are only asked how many times the graph cuts the axis, the count of distinct zeros is the count of crossing points.
04

Zeros of a Quadratic by Factorisation

The method is the same every time: make the left-hand side zero, then split it into two brackets, then set each bracket to zero in turn. The whole skill is spotting the two numbers whose product is the constant term and whose sum is the coefficient of x, with the sign moved across.

  • x² − 5x + 6 = 0. We need −5 as the sum and +6 as the product: that is −2 and −3. So (x − 2)(x − 3) = 0, giving x = 2 or x = 3.
  • x² + x − 12 = 0. We need +1 as the sum and −12 as the product: that is +4 and −3. So (x + 4)(x − 3) = 0, giving x = −4 or x = 3.
  • 2x² − 3x − 5 = 0. Split as (2x − 5)(x + 1) = 0, since 2x² + 2x − 5x − 5 = 2x² − 3x − 5. So 2x − 5 = 0 or x + 1 = 0, giving x = 5/2 or x = −1.
  • 3x² − 5x + 2 = 0. Split as (3x − 2)(x − 1) = 0, since 3x² − 3x − 2x + 2 = 3x² − 5x + 2. So x = 2/3 or x = 1.
Factorisation turns one quadratic into two linear equations.

Checking the split in one line

Expand the brackets back and confirm the middle term before you solve. (x − 2)(x − 3) gives x² − 5x + 6, which matches. (3x − 2)(x − 1) gives 3x² − 5x + 2, which also matches. Thirty seconds of checking removes almost every wrong answer in this question type, because the zeros are the whole of the answer and a wrong bracket destroys both of them.
05

Zeros and Coefficients: The Two Relations

This is the highest-value item in the chapter. If the zeros of p(x) = ax² + bx + c are α and β, then the coefficients of the polynomial are fixed by the zeros alone. The sum of the zeros is −b/a and the product of the zeros is c/a. Read them as instructions for building a polynomial: to get a quadratic with known zeros, write down the sum, put a minus sign on it, and use it as the coefficient of x.

The two relations between the zeros α, β and the coefficients a, b, c.
  • x² − 5x + 6 = 0 with zeros 2 and 3: the sum is 5 and −b/a = 5, so −b/a checks. The product is 6 and c/a = 6, so c/a checks.
  • 2x² − 3x − 5 = 0 with zeros 5/2 and −1: the sum is 5/2 − 1 = 3/2, and −b/a = 3/2. The product is −5/2, and c/a = −5/2.
  • x² + x − 12 = 0 with zeros 3 and −4: the sum is −1, and −b/a = −1. The product is −12, and c/a = −12.
  • These two relations are also the test of a quadratic: a pair of proposed zeros is right only if both the sum and the product come out right.

Where students lose the mark

The sign on the sum. It is −b/a and not b/a, because α + β = −5 for x² − 5x + 6, not +5. And the product is c/a with no sign change at all, so for x² + x − 12 the product −12 is correct while the sum is −1. Write the two relations out on the first line of your answer and copy them faithfully.
06

Building a Polynomial from Its Zeros

The two relations run backwards without any change. If a quadratic has zeros α and β then it must be a constant multiple of (x − α)(x − β), and with leading coefficient a that multiple is exactly a. So a quadratic with known zeros is a one-line answer, and this is the usual five-mark question: find the quadratic, then check it.

The polynomial with given zeros α and β and leading coefficient a.
  • Find the quadratic with zeros 3 and −4 and leading coefficient 1: p(x) = (x − 3)(x + 4) = x² + x − 12.
  • Find the quadratic with zeros 3 and −4 and leading coefficient 2: p(x) = 2(x − 3)(x + 4) = 2x² + 2x − 24.
  • Check the second one: α + β = −1 = −2/2, and αβ = −12 = −24/2. Both relations hold.
  • If the leading coefficient is not given, the answer is not unique. Any non-zero multiple of the same polynomial has the same zeros, so state that the leading coefficient has been taken as 1.

Finding a missing value from the zeros

If the zeros of x² + kx + 6 are 2 and 3, then the sum is 5, so −k = 5 and k = −5. Then the polynomial is x² − 5x + 6, and the product check confirms it: 2 × 3 = 6, which is the constant term. Always finish with the product check, because the sum alone will not always pin the value down.
07

How the Questions Are Asked

Unit II Algebra carries 20 of the 80 written marks and Polynomials is the first section of it. The questions are short, structured and repetitive, so a fixed method earns the marks every time: state p(x) = 0, factorise, list the zeros, and then apply whichever relation the question asks for.

  • Find the zeros of the polynomial x² − 5x + 6 and verify the relation between the zeros and the coefficients. (3 marks, Section C.)
  • Find the zeros of 2x² − 3x − 5 by factorisation. (2 marks, Section B.)
  • If the zeros of the polynomial x² + kx + 6 are 2 and 3, find the value of k. (2 marks, Section B.)
  • Find the quadratic polynomial whose zeros are 3 and −4 and whose leading coefficient is 2. (5 marks, Section D.)
  • Sketch the graph of y = x² − 5x + 6 and mark its zeros on it. (3 marks, Section C.)
  • A number k is a zero of the polynomial p(x) = 3x² − 5x + 2. Find all possible values of k. (1 to 2 marks.)

The value that catches everyone

A zero k of p(x) satisfies p(k) = 0, so p(k) = 3k² − 5k + 2 = 0 gives (3k − 2)(k − 1) = 0 and k = 2/3 or k = 1. Students read the question as p(x) = k and lose the mark. Write p(k) = 0 on the first line every time.

Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Definition of a zero

A zero is a value of x, not of y. Every zeros question starts with p(x) = 0.

Sum of the zeros of a quadratic

The sum carries the minus sign; this is where the mark is most often lost.

Product of the zeros of a quadratic

The product has no sign change at all.

Quadratic from its zeros

The reverse direction of the two relations.

Geometric meaning of a zero

A zero is an x-coordinate, so two distinct zeros mean two crossings of the x-axis.

Factoring a quadratic

Find two numbers with the right product and the right sum, then write the brackets.

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

  • A zero k satisfies p(k) = 0. Write p(k) = 0 on the first line; students who read it as p(x) = k lose the mark before they start.
  • Sum of zeros is −b/a, product of zeros is c/a. The minus sign belongs to the sum only.
  • A polynomial of degree n has at most n zeros, so a quadratic gives at most two and a linear polynomial gives at most one.
  • A zero is the x-coordinate of an intersection with the x-axis: two distinct zeros cut the axis twice, one repeated zero means the graph touches and turns back.
  • Factorisation method: put p(x) = 0, find two numbers with product = constant term and sum = the middle coefficient, write the brackets, then set each bracket to zero.
  • If the leading coefficient is not given, the polynomial is not unique — any non-zero multiple has the same zeros, so say that you have taken the leading coefficient as 1.
  • Do not use the division algorithm for polynomials. It was removed from the 2024-25 course and is not needed for any of the above.
  • Unit II Algebra carries 20 of the 80 written marks; the remaining 20 marks are internal assessment, 10 for the pen-paper test and multiple assessment, 5 for the portfolio and 5 for the lab practical.

FAQ

Frequently asked questions

What is a zero of a polynomial?

A real number k is a zero of the polynomial p(x) if p(k) = 0. Finding the zeros of p is therefore the same work as putting p(x) = 0 and solving that equation. For p(x) = x² − 5x + 6, the equation is (x − 2)(x − 3) = 0, so the zeros are 2 and 3. The point (k, 0) is called the zero point of the graph.

What is the relationship between the zeros and the coefficients of a quadratic polynomial?

If the zeros of p(x) = ax² + bx + c are α and β, then α + β = −b/a and αβ = c/a. The sum of the zeros equals the coefficient of x with its sign changed, divided by the leading coefficient; the product of the zeros equals the constant term divided by the leading coefficient, with no sign change.

How many zeros can a polynomial of degree n have?

At most n. A linear polynomial has at most one zero, and a quadratic has at most two. If the two values come out equal, the quadratic has one zero repeated twice rather than two separate zeros, which is why the graph of x² + 4x + 4 touches the x-axis at (−2, 0) instead of cutting it.

How do you find a quadratic polynomial when the zeros are given?

Multiply the leading coefficient by the two brackets. With zeros α and β, p(x) = a(x − α)(x − β), which expands to a[x² − (α + β)x + αβ]. So for zeros 3 and −4 with leading coefficient 2, p(x) = 2(x − 3)(x + 4) = 2x² + 2x − 24. If no leading coefficient is given, the answer is only determined up to a non-zero constant multiple.

What does a zero look like on the graph of a polynomial?

It is the x-coordinate of a point where the graph meets the x-axis. The graph of y = p(x) cuts the x-axis at (k, 0) for each zero k, because at that point p(k) = 0. So the zeros of x² − 5x + 6, namely 2 and 3, appear as the points (2, 0) and (3, 0).

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