Polynomials Class 10 Most Important Questions for Boards

Polynomials (Class 10 Mathematics)

Introduction

English

A Polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication. Polynomials are one of the most important topics in Class 10 Mathematics and form the foundation for higher algebra. ( Polynomials Class 10 )

हिंदी

Polynomial (बहुपद) एक बीजीय व्यंजक (Algebraic Expression) होता है जिसमें चर (Variables) और गुणांक (Coefficients) जोड़, घटाव तथा गुणा के माध्यम से जुड़े होते हैं। यह अध्याय आगे की Algebra तथा Competitive Exams की मजबूत नींव तैयार करता है।

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Table of Contents ( Polynomials Class 10 )

1. What is Polynomial?
2. Types of Polynomials
3. Degree of Polynomial
4. Zeroes of Polynomial
5. Relationship Between Zeroes and Coefficients
6. Division Algorithm
7. Important Formulas
8. NCERT Important Questions
9. PYQs
10. HOTS Questions
11. Key Points
12. Practice Questions for Readers

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Abbreviations

Abbreviation	Full Form
PYQ	Previous Year Question
HOTS	Higher Order Thinking Skills
MCQ	Multiple Choice Question
NCERT	National Council of Educational Research and Training
LHS	Left Hand Side
RHS	Right Hand Side

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What is a Polynomial?

English

An expression of the form:

$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$an​xn+an−1​xn−1+⋯+a1​x+a0​

where coefficients are real numbers and powers of variables are non-negative integers.

हिंदी

जिस बीजीय व्यंजक में चर की घात (Power) शून्य या धनात्मक पूर्णांक हो, उसे Polynomial कहते हैं।

Examples

• x + 2
• x² + 5x + 6
• 3x³ – 2x² + 4

Not Polynomial

• 1/x
• √x
• x⁻²

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Types of Polynomials

Based on Number of Terms

Terms	Polynomial
One Term	Monomial
Two Terms	Binomial
Three Terms	Trinomial

Examples

• 5x → Monomial
• x + 3 → Binomial
• x² + 4x + 7 → Trinomial

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Based on Degree

Degree	Polynomial
0	Constant
1	Linear
2	Quadratic
3	Cubic

Examples

• 5 → Constant
• x + 1 → Linear
• x² + 3x + 2 → Quadratic
• x³ + 2x² + x → Cubic

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Degree of Polynomial

English

The highest power of the variable in a polynomial is called its degree.

हिंदी

किसी Polynomial में चर की सबसे बड़ी घात को उसकी Degree कहते हैं।

Example

Polynomial:

x³ + 2x² + 5

Degree = 3

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Zeroes of a Polynomial

English

A value of x that makes the polynomial equal to zero is called a zero of the polynomial.

हिंदी

Polynomial को शून्य करने वाले x के मान को Zero of Polynomial कहते हैं।

Example

For:

$p(x)=x^2-5x+6$p(x)=x2−5x+6

Factorization:

(x − 2)(x − 3)

Therefore,

Zeroes = 2, 3

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Relationship Between Zeroes and Coefficients

For quadratic polynomial:

$ax^2+bx+c$ax2+bx+c

If α and β are zeroes,

Sum of Zeroes

$\alpha+\beta=-\frac{b}{a}$α+β=−ab​

Product of Zeroes

$\alpha\beta=\frac{c}{a}$αβ=ac​

Example

x² – 7x + 12

a = 1, b = -7, c = 12

Sum = 7

Product = 12

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Division Algorithm

Statement

Dividend = Divisor × Quotient + Remainder

Formula

$p(x)=g(x)q(x)+r(x)$p(x)=g(x)q(x)+r(x)

हिंदी

किसी Polynomial को दूसरे Polynomial से भाग देने पर प्राप्त नियम को Division Algorithm कहते हैं।

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Important Exam Formulas ( Polynomials Class 10 )

Formula	Use
α + β = -b/a	Sum of Zeroes
αβ = c/a	Product of Zeroes
Dividend = Divisor × Quotient + Remainder	Division Algorithm
Degree of Constant Polynomial = 0	Concept

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NCERT Important Questions with Solutions ( Polynomials Class 10 )

Question 1

Find the zeroes of:

x² − 9

Solution

x² − 9 = (x + 3)(x − 3)

Zeroes:

• 3
• -3

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Question 2

Find the sum and product of zeroes of:

x² + 6x + 8

Solution

a = 1

b = 6

c = 8

Sum = -6

Product = 8

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Question 3

Verify relationship between zeroes and coefficients of:

x² − 5x + 6

Solution

Zeroes = 2 and 3

Sum = 5

Product = 6

Verified.

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Previous Year Questions (PYQs) ( Polynomials Class 10 )

PYQ 1

Find the zeroes of:

x² − 4x + 4

Answer

( x − 2 )²

Zeroes = 2, 2

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PYQ 2

Find the remainder when:

x³ − 5x² + 3x + 1

is divided by (x − 1)

Solution

Put x = 1

Remainder = 0

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PYQ 3

If α and β are zeroes of x² − 8x + 15, find α + β and αβ.

Answer

α + β = 8

αβ = 15

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HOTS Questions

HOTS 1

If one zero of polynomial x² − 9x + k is 3, find k.

Solution

3² − 9(3) + k = 0

9 − 27 + k = 0

k = 18

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HOTS 2

Find a quadratic polynomial whose zeroes are 4 and -2.

Solution

Required Polynomial

(x − 4)(x + 2)

= x² − 2x − 8

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HOTS 3

The sum of zeroes is 7 and product is 12. Form the polynomial.

Solution

x² − 7x + 12

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Important Key Points

Polynomial में Variable की Power Negative नहीं होती।

Degree = Highest Power.

Quadratic Polynomial के अधिकतम 2 Zeroes होते हैं।

Cubic Polynomial के अधिकतम 3 Zeroes होते हैं।

Graph x-axis को जहां काटता है वही Zeroes होते हैं।

Sum of Zeroes = -b/a

Product of Zeroes = c/a

Board Exam में Relationship Between Zeroes and Coefficients से प्रश्न अक्सर पूछे जाते हैं।

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Most Important MCQs

1. Degree of polynomial x³ + x² + 5 is

A) 1

B) 2

C) 3

D) 5

Answer: C

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2. Zeroes of x² − 25 are

A) ±5

B) ±4

C) ±3

D) ±2

Answer: A

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3. Product of zeroes of x² + 4x + 3 is

A) 3

B) 4

C) -4

D) -3

Answer: A

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Reader Challenge Zone 🏆

Solve These 5 Questions

Q1.

Find the zeroes of:

x² − 11x + 30

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Q2.

Find the sum and product of zeroes of:

x² − 9x + 20

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Q3.

Form a quadratic polynomial whose zeroes are 5 and 7.

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Q4.

Find the remainder when:

x³ + 2x² − 5x + 7

is divided by (x − 2).

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Q5.

If one zero of x² − 6x + k is 2, find k.

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📢 Viewer Participation Section

English:
Students who submit all 5 correct answers through the comment section or email will have their Name, City/Location, and Score featured in the next chapter’s “Top Learners Wall of Fame” section on Rising Star Mindset.

हिंदी:
जो विद्यार्थी ऊपर दिए गए सभी 5 प्रश्नों के सही उत्तर Comment या Email के माध्यम से भेजेंगे, उनका नाम, स्थान (Location) और Score अगले Chapter के “Top Learners Wall of Fame” सेक्शन में प्रकाशित किया जाएगा।

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Conclusion

Polynomials is one of the most important chapters of Class 10 Mathematics. A strong understanding of zeroes, degree, factorization, and the relationship between zeroes and coefficients will help students score high marks in Board Exams and build a solid foundation for higher mathematics.

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www.risingstarmindset.com

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