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.
Key Concepts
Concepts grouped the way the chapter is taught — open the bucket that matches what you want to revise.
Core Concepts
high priorityOpen the chapter concepts in a clean revision order.
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.
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 ax^2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.
Cubic polynomial and its zeros
A cubic polynomial is a polynomial of degree 3, usually written as ax^3+bx^2+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 quadratic polynomial
If α and β are the zeros of the quadratic polynomial ax^2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.
Relationship between zeros and coefficients of a cubic polynomial
If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.
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.
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.
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.
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.
Exam Intelligence
Use this section to decide what deserves the most revision time.
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.
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, or conditions
- Short answer: worked method, example, or reason-based explanation
- Case-based: chapter scenario with concept-linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- Zero means the value of x that makes the polynomial zero.
- Linear polynomials have one zero; quadratic polynomials have at most two; cubic polynomials have at most three.
- Sum and product relations are powerful shortcuts for quadratic and cubic polynomials.
- Factorisation, verification, and polynomial division are the main exam tools in this chapter.
- Graph reading helps connect algebraic zeros with visible x-intercepts and sign changes.
- 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 o…
- 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 ax^2+bx+c where a is not zero. Its zeros are the values of x that ma…
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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