Chapter Hub
Polynomials
This chapter builds the idea that a polynomial is not only an algebraic expression but also a graph with useful meaning. Students learn how zeros, coefficients, and graphs are connected, and how to use these ideas in direct questions and case-based problems. For Class 10 exams, the chapter is important because questions often test zeros of polynomials, relationships between zeros and coefficients, factorisation, verification, division algorithm, and interpretation of the graph. A clear method and neat algebra are usually enough to score full marks.
Difficulty
Medium
Study time
96-120 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 96 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Zeros and Graph Meaning
Covers Geometrical meaning of zero of a polynomial and Sign pattern and graph interpretation.
Open mapLinear and Quadratic Basics
Covers Linear polynomial and its zero, Quadratic polynomial and its zeros, Relationship between zeros and coefficients of a quadratic polynomial and Forming quadratic polynomial from given zeros.
Open mapCubic Polynomial Skills
Covers Cubic polynomial and its zeros, Relationship between zeros and coefficients of a cubic polynomial and Forming cubic polynomial from given zeros.
Open mapChecking and Division
Covers Verification of given zeros, Division algorithm for polynomials and Checking remainder through factor.
Open mapKey Concepts
Concepts grouped the way the chapter is taught — open the group that matches what you want to revise.
Zeros and Graph Meaning
Covers Geometrical meaning of zero of a polynomial and Sign pattern and graph interpretation.
Geometrical meaning of zero of a polynomial
A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.
Sign pattern and graph interpretation
The sign pattern of a polynomial tells whether the graph lies above or below the x-axis in different intervals. The x-intercepts show the zeros, and the shape of the graph helps interpret the number of real zeros.
Linear and Quadratic Basics
Covers Linear polynomial and its zero, Quadratic polynomial and its zeros, Relationship between zeros and coefficients of a quadratic polynomial and Forming quadratic polynomial from given zeros.
Linear polynomial and its zero
A linear polynomial has highest power 1, usually written as ax+b where a is not zero. Its zero is the value of x that makes ax+b equal to 0.
Quadratic polynomial and its zeros
A quadratic polynomial is a polynomial of degree 2, usually written as ax2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.
Relationship between zeros and coefficients of a quadratic polynomial
If α and β are the zeros of the quadratic polynomial ax2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.
Forming quadratic polynomial from given zeros
If α and β are the zeros of a quadratic polynomial, then the polynomial can be written as k(x-α)(x-β), where k is a non-zero constant.
Cubic Polynomial Skills
Covers Cubic polynomial and its zeros, Relationship between zeros and coefficients of a cubic polynomial and Forming cubic polynomial from given zeros.
Cubic polynomial and its zeros
A cubic polynomial is a polynomial of degree 3, usually written as ax3+bx2+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.
Relationship between zeros and coefficients of a cubic polynomial
If α, β, and γ are the zeros of ax3+bx2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.
Forming cubic polynomial from given zeros
If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant.
Checking and Division
Covers Verification of given zeros, Division algorithm for polynomials and Checking remainder through factor.
Verification of given zeros
To verify whether a number is a zero of a polynomial, substitute that number into the polynomial and check whether the value becomes 0.
Division algorithm for polynomials
For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x).
Checking remainder through factor
If a polynomial p(x) is divided by x-a, then the remainder is p(a). If p(a)=0, then x-a is a factor of the polynomial.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
High Probability Topics
- Geometrical meaning of zero of a polynomial
- Linear polynomial and its zero
- Quadratic polynomial and its zeros
- Cubic polynomial and its zeros
- Relationship between zeros and coefficients of a quadratic polynomial
- Relationship between zeros and coefficients of a cubic polynomial
- Forming quadratic polynomial from given zeros
- Forming cubic polynomial from given zeros
Common Traps
- Using the wrong sign in factor form x-a.
- Forgetting the minus sign in the sum of zeros of a quadratic.
- Stopping after only partial factorisation of a cubic polynomial.
- Treating y-coordinate as the zero instead of x-coordinate.
- Using a remainder whose degree is not smaller than the divisor.
How CBSE Asks This Chapter
- Geometrical meaning of zero of a polynomial
Find the zeros from an expression or from a graph, and state their geometrical meaning in one line.
- Linear polynomial and its zero
Find the zero of a given linear polynomial, or check whether a stated value is a zero.
- Quadratic polynomial and its zeros
Find the zeros of a quadratic polynomial, verify them, or relate them to the graph.
- Cubic polynomial and its zeros
Find the zeros of a cubic polynomial, or use factorisation to verify a stated root.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Geometrical meaning of zero of a polynomial
Remember
If p(a)=0, then a is a zero of the polynomial p(x).Trap to avoid
Many students think a zero means the y-value is zero for every point.
Linear polynomial and its zero
Remember
For p(x)=ax+b, a zero is x=-b/a, with a≠0.Trap to avoid
Students often change the sign wrongly while moving the constant term.
Quadratic polynomial and its zeros
Remember
For ax2+bx+c, zeros are the x-values satisfying ax2+bx+c=0.Trap to avoid
Students sometimes think a quadratic must always have two different zeros.
Cubic polynomial and its zeros
Remember
For ax3+bx2+cx+d, the number of zeros is at most 3.Trap to avoid
Students sometimes stop after finding two zeros and forget that a cubic polynomial may have three zeros.
Relationship between zeros and coefficients of a quadratic polynomial
Remember
For a quadratic polynomial ax2+bx+c with zeros α and β: α+β=-b/a and αβ=c/a.Trap to avoid
Students often forget the minus sign in the sum formula or write product as -c/a.
Relationship between zeros and coefficients of a cubic polynomial
Remember
For zeros α, β, γ of ax3+bx2+cx+d: α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.Trap to avoid
Students often forget that the product of three zeros has a negative sign with d/a in the standard form ax3+bx2+cx+d.
Forming quadratic polynomial from given zeros
Remember
Polynomial with zeros α and β: k(x-α)(x-β), k≠0.Trap to avoid
Students sometimes write (x+α)(x+β) even when the zeros are positive numbers.
Forming cubic polynomial from given zeros
Remember
Cubic with zeros α, β, γ: k(x-α)(x-β)(x-γ), k≠0.Trap to avoid
Students sometimes forget one zero and write only a quadratic polynomial.
Remember
A number a is a zero of p(x) if and only if p(a)=0.Trap to avoid
Students often substitute only one part of the expression and forget to use the full polynomial.
Division algorithm for polynomials
Remember
p(x)=g(x)q(x)+r(x), with degree of r(x) < degree of g(x).Trap to avoid
Students sometimes write a remainder whose degree is equal to or greater than the divisor.
Checking remainder through factor
Remember
Remainder on division by x-a is p(a).Trap to avoid
Students often reverse the factor sign and test the wrong value.
Sign pattern and graph interpretation
Remember
Above x-axis implies p(x)>0, on x-axis implies p(x)=0, and below x-axis implies p(x)<0.Trap to avoid
Students sometimes think every polynomial must have the same sign on both sides of the x-axis.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Geometrical meaning of zero of a polynomial | If p(a)=0, then a is a zero of the polynomial p(x). | Many students think a zero means the y-value is zero for every point. |
| Linear polynomial and its zero | For p(x)=ax+b, a zero is x=-b/a, with a≠0. | Students often change the sign wrongly while moving the constant term. |
| Quadratic polynomial and its zeros | For ax2+bx+c, zeros are the x-values satisfying ax2+bx+c=0. | Students sometimes think a quadratic must always have two different zeros. |
| Cubic polynomial and its zeros | For ax3+bx2+cx+d, the number of zeros is at most 3. | Students sometimes stop after finding two zeros and forget that a cubic polynomial may have three zeros. |
| Relationship between zeros and coefficients of a quadratic polynomial | For a quadratic polynomial ax2+bx+c with zeros α and β: α+β=-b/a and αβ=c/a. | Students often forget the minus sign in the sum formula or write product as -c/a. |
| Relationship between zeros and coefficients of a cubic polynomial | For zeros α, β, γ of ax3+bx2+cx+d: α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a. | Students often forget that the product of three zeros has a negative sign with d/a in the standard form ax3+bx2+cx+d. |
| Forming quadratic polynomial from given zeros | Polynomial with zeros α and β: k(x-α)(x-β), k≠0. | Students sometimes write (x+α)(x+β) even when the zeros are positive numbers. |
| Forming cubic polynomial from given zeros | Cubic with zeros α, β, γ: k(x-α)(x-β)(x-γ), k≠0. | Students sometimes forget one zero and write only a quadratic polynomial. |
| Verification of given zeros | A number a is a zero of p(x) if and only if p(a)=0. | Students often substitute only one part of the expression and forget to use the full polynomial. |
| Division algorithm for polynomials | p(x)=g(x)q(x)+r(x), with degree of r(x) < degree of g(x). | Students sometimes write a remainder whose degree is equal to or greater than the divisor. |
| Checking remainder through factor | Remainder on division by x-a is p(a). | Students often reverse the factor sign and test the wrong value. |
| Sign pattern and graph interpretation | Above x-axis implies p(x)>0, on x-axis implies p(x)=0, and below x-axis implies p(x)<0. | Students sometimes think every polynomial must have the same sign on both sides of the x-axis. |
Practice
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