Class 10 Maths Chapter 2 Polynomials Important Questions with Solutions (CBSE 2026 PDF)

Hello students…. today i am going to find Class 10 Maths Chapter 2 Polynomials Important Questions, As you move further into your Class 10 , you’ll find that Algebra forms the backbone of your Mathematics syllabus.

For your 2026 Board Exams, this chapter is a scoring. Whether it’s finding the “zeroes” of a quadratic equation or understanding the hidden relationship between those zeroes and their coefficients, getting these concepts right early on will give you a massive advantage.

In this article, we have solved the list of Class 10 Maths Chapter 2 Polynomials Important Questions. We’ve focused on the latest CBSE patterns to ensure you are practicing exactly what matters. From graphical interpretations to algebraic verifications, these notes are designed to turn your confusion into confidence.

Introduction to Class 10 Maths Chapter 2 Polynomials

In the 2026 examination architecture, Polynomials is not just a standalone chapter; it is a high-yield section within the Algebra Unit, which carries the highest weightage in the entire paper.

Marks Distribution

• Unit Weightage: Algebra (including Polynomials, Linear Equations, Quadratic Equations, and AP) carries 20 Marks out of 80.
• Chapter Specifics: Expect roughly 4 to 6 marks directly from Polynomials. It usually appears as a combination of 1-mark MCQs and a 3-mark Short Answer question.

Deleted Syllabus

Students often waste time on topics no longer in the curriculum. For the 2026 exam, the Division Algorithm for Polynomials (Long Division) and related complex problems remain deleted. Focus your energy entirely on the relationship between zeroes and coefficients.

Competency

For 2026, CBSE has increased Competency-Based Questions (CBQs) to 50% of the paper. For Polynomials, this means:

• Case Study Potential: Polynomials are frequently used for Case-Based questions (Section E). You might see a real-life scenario, like the path of a projectile or a bridge’s arch, where students must identify the polynomial type and its zeroes.
• Conceptual Depth: Gone are the days of simple rote memorization. The 2026 paper will test why a graph has no zeroes or how changing a coefficient shifts the entire curve.

What Are Polynomials? Basic Concept Explained

In simple words, a Polynomial is like a “chain” of mathematical terms. It is an algebraic expression made up of variables (like x or y), exponents, and numbers, all tied together by addition or subtraction.

The only rule? The power (exponent) of the variable must be a whole number (0, 1, 2, 3…). If you see a square root on x or x in the denominator, it’s not a polynomial!

Terms

Terms is very simple meaning think of terms as the individual “bricks” that build the expression. They are separated by + or – signs.

Example: In $3x^2$ + 5x – 7, there are three terms: $3x^2$, 5x, and -7

Coefficients

These are the “numbers” standing right next to the variables. They multiply the variable.

Example: In $4x^2$ – 2x + 9

• The coefficient of$x^2$ is 4.
• The coefficient of x is -2
• The 9 is a constant because it has no variable attached.

Degree

The degree is the “power level” of the polynomial. It is simply the highest power of the variable in the entire expression. It tells you how many zeroes (solutions) the polynomial can have.

Expression	Terms	Leading Coefficient	Degree	Name
5x + 3	5x, 3	5	1	Linear
$2x^2$ – 3x + 4	$2x^2$, -3x, 4	2	2	Quadratic
$x^3$ – 1	$x^3$, -1	1	3	Cubic

Types of Polynomials Based on Degree

Here is the breakdown of polynomials based on their Degree

1. Zero Polynomial
2. Constant Polynomial
3. Linear Polynomial
4. Quadratic Polynomial
5. Cubic Polynomial

Name	Degree	General Form	Max Zeroes
Constant	0	k	0
Linear	1	ax + b	1
Quadratic	2	$ax^2$ + bx + c	2
Cubic	3	$ax^3 + bx^2 + cx + d$	3

Types of Polynomials Based on Terms

Monomial

A monomial is a polynomial that has only one term.

Binomial

A binomial has exactly two terms separated by a plus or minus sign.

Trinomial

A trinomial has exactly three terms.

In Class 10, most of the Quadratic Polynomials you solve will be trinomials.

Name	Number of Terms	Examples
Monomial	1	7x, -12, $x^3$
Binomial	2	x – 9, $4x^2$ + 1
Trinomial	3	$4x^2$ + bx + c, $x^2$ + 2x + 1
Polynomial	Many	$x^4 + 3x^3 – 2x + 5$

Key Topics Covered in Class 10 Maths Chapter 2 Polynomials

There is key topics of Class 10 Maths Chapter 2 Polynomials Important Questions 2026

• Zeroes of a Polynomial
• Zeroes of a Polynomial
• Relationship Between Zeroes and Coefficients
• Forming a Quadratic Polynomial

Concept	Expected Marks	Difficulty Level
Identifying zeroes from a graph	1 Mark	Easy
Relationship Verification (alpha, beta)	2-3 Marks	Moderate
Forming an equation from roots	2 Marks	Easy
Case Study (Parabola applications)	4 Marks	Hard

Zeros of a Polynomial

This is the most “visual” part of the chapter. If you can understand the graph, you can solve the first 2 or 3 questions of any board paper in seconds.

A Zero is a “magic number” that you plug into x to make the whole polynomial equal to 0.

• If p(x) = x – 3, the zero is 3 because 3 – 3 = 0
• If p(x) = $x^2$ – 4, the zeroes are 2 and -2 because both $2^2$– 4 and $(-2)^2$ – 4 = 0

The Graphical Meaning

You don’t always need to solve the equation to find the number of zeroes. You can just look at its graph

The number of zeroes is exactly equal to the number of times the graph intersects (touches or crosses) the x-axis

1. We ignore the y-axis entirely for this.
2. If the graph crosses the x-axis at 3 points, it has 3 zeroes.
3. If it just touches and bounces back, that counts as one point (two equal zeroes).
4. If it never touches the x-axis, it has no real zeroes.

The Parabola (Quadratic Polynomials)

In Class 10, we focus heavily on the graph of ax^2 + bx + c, which is a U-shaped curve called a Parabola. There are three main cases you need to know

Case	Graph Description	Number of Zeroes
Case 1	Crosses x-axis at two distinct points.	2 Zeroes
Case 2	Touches x-axis at exactly one point.	1 Zero (Two equal zeroes)
Case 3	Completely above or below the $x$-axis.	0 Real Zeroes

If a > 0 (Positive): The parabola opens Upwards (like a smiley face up).

If a < 0 (Negative): The parabola opens Downwards (like a sad face down).

Relationship Between Zeros and Coefficients

There is a relation between Zeros and coefficients if you use formula and factorise and you got two answers , i mean solution they are zeros and verified with sum of zero and product of zeros formula. in the pdf file of Class 10 Maths Chapter 2 Polynomials Important Questions i have solved step by step of relation between Zeros and coefficients.

Class 10 Maths Chapter 2 Polynomials Important Questions with Solutions

Top Exam-Oriented Questions: Chapter 2 – Polynomials

1 Mark Questions from Polynomials

1. If the graph of a polynomial p(x) does not cut the x-axis at any point, what is the number of zeroes of p(x)?
2. If the sum of the zeroes of the quadratic polynomial $3x^2 – kx + 6$ is 3, find the value of k.
3. Can a quadratic polynomial have more than 2 zeroes?
4. Write a quadratic polynomial whose zeroes are 2 and -3.
5. If the product of the zeroes of a$x^2 – 6x – 6$ is 4, find the value of a.
6. Is p(x) = $x^2 + 2 \sqrt x + 5$a polynomial? Why?
7. What is the shape of the graph of a quadratic polynomial?

2 Mark Important Questions

1. Find the zeroes of $4x^2 – 4x + 1$ and verify the relationship.
2. If $\alpha$ and $\beta$ are zeroes of $x^2 – 5x + 6$, find the value of $\alpha^2 + \beta^2$.
3. If one zero of the polynomial $(k-1)x^2 + kx + 1$ is -3, then find the value of k.
4. Find the zeroes of $x^2 – 3$ and verify the relationship.

3 Mark Important Questions

1. If$\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = 3x^2 – 4x + 1$, find a quadratic polynomial whose zeroes are $\frac{\alpha^2}{\beta}$ and $\frac{\beta^2}{\alpha}$.
2. If the zeroes of the polynomial $x^2 – px + q$ are double in value to the zeroes of $2x^2 – 5x – 3$, find the values of p and q.
3. f $\alpha$ and $\beta$ are zeroes of $x^2 – k(x + 1) – c$ such that $(\alpha + 1)(\beta + 1) = 0$, then find the value of c
4. Find the zeroes of the quadratic polynomial $4\sqrt{3}x^2 + 5x – 2\sqrt{3}$

4 Mark Important Questions

HOTS Questions (4 Marks) – Class 10 Polynomials

1. A polynomial $p(x)$p(x) leaves a remainder 2 when divided by $(x-1)$(x−1) and a remainder -1 when divided by $(x+1)$(x+1).
2. If $x+2$x+2 and $x-3$x−3 are factors of the polynomial $p(x)=x^3+ax^2+bx−6$ . Find the values of a and b.
3. The polynomial $f(x)=2x^3−7x^2−5x+6$ has a factor $(x-3)$(x−3). Find the other factors of the polynomial.
4. If the sum of zeroes of a quadratic polynomial is 6 and the product is 8, construct the polynomial.
5. If α and β are the zeroes of the polynomial$x^2−5x+6$ , Find the polynomial whose zeroes are $(α+1),(β+1)$
6. If α and β are the zeroes of the polynomial $2x^2−7x+3$ , Find the value of $α2+β2$.
7. Find the remainder when $p(x)$p(x) is divided by $(x-1)(x+1)$(x−1)(x+1).

Step-by-Step Solutions to Important Polynomial Questions

Here you got the Step-by-Step Solutions of Class 10 Maths Chapter 2 Polynomials Important Questions.

1 mark solution from Polynomials

1. The number of zeroes is 0.

(Zeroes of a polynomial are the x-coordinates where the graph cuts the x-axis.)

2.For a quadratic $ax^2 + bx + c$

$$\alpha + \beta = -\frac{b}{a}$$

Here

$$a=3 , b=−k$$

$$α+β=−(-k/3)= k/3$$

$$k/3​=3$$

$$k=9 \\\\\\Answer$$

3. No. A quadratic polynomial has at most 2 zeroes.

4. Polynomial with zeroes $\alpha, \beta$

Answer:$x^2 + x – 6$

5. For quadratic polynomial

Here $c = -6$

6. No. Because in a polynomial, the powers of the variable must be non-negative integers, but here the term $\sqrt{x} = x^{1/2}$x​=$x^1/2$ has a fractional power.

7. The graph of a quadratic polynomial is a Parabola.

2 Mark Important Questions

Solution 1

Step 1 factorise the given polynomial

then x= 1/2 , 1/2

Step 2 Verify the relationship

For a quadratic polynomial

Sum of zeroes

hence proved the relationship between zeroes and coefficients.

Solution 2

We have given the Polynomal $x^2-5x+6$

compare the equation $ax^2+bx+c=0$ ,then we have a=1 ,b=-5 , c=6

now $\alpha+\beta= \frac{-b}{a}$

now factorise the same polynomial $x^2-5x+6$

let $\alpha=2$ and $\beta$=3 now check $\alpha+\beta=2+3=5$

and $\alpha.\beta=2$x3 = 6

Hence proved the relationship between zeroes and coefficients.

Solution 3

Given the polynomial

and -3 is one zero ,, so putting x=-3 in above equation.

Required value of k is 4/3.

Solution 4

given

zeros are $+\sqrt3,-\sqrt3$

After comparing the standard equation we get a=1,b=0,c=-3

Hence proved the relationship between zeroes and coefficients.

Note – 3 marks and 4 marks ke solution ke liye aap humara telegram join kar lijiye or ha top 20 questions pdf file jisme Class 10 Maths Chapter 2 Polynomials Important Questions with Solutions step by step hai meri handwriting me use jarur download kare.

Frequently Asked Board Questions

(CBSE Class 10 Maths Chapter 2 – Polynomials)

1. Find the zeroes of the quadratic polynomial $2x^2−7x+3$and verify the relationship between zeroes and coefficients.
2. Find the value of $k$k so that $x+1$ is a factor of $2x^3+kx^2−x−1$
3. If the sum and product of zeroes of a quadratic polynomial are 5 and 6 respectively, find the polynomial.
4. Find the remainder when $x^3 – 4x^2 + 3x – 5$ is divided by $(x-2)$.
5. Check whether $(x-1)$(x−1) is a factor of the polynomial $x^3−3x^2+3x−1$.
6. Find the cubic polynomial whose zeroes are $1, -1, 2$.
7. If one zero of the polynomial $x^2 – 5x + k$ is 2, find the value of $k$
8. Find the number of zeroes of the polynomial from its graph. (Graph-based question – very common in boards)
9. Divide the polynomial $2x^3 + 3x^2 – 5x + 6$ by $(x-2)$ using polynomial division.
10. Find the value of a such that $(x+2)$(x+2) is a factor of $x^3+ax^2−x−2$
11. Find the quadratic polynomial whose zeroes are $\frac{1}{2}$ and $-3$.
12. Find the remainder when a polynomial is divided by a linear polynomial using the Remainder Theorem.
13. Verify whether the given polynomial satisfies the Factor Theorem.
14. Find the relationship between zeroes and coefficients of a quadratic polynomial and verify it for a given polynomial.
15. Construct a polynomial whose graph has zeroes at given points.

Tips to Solve Polynomial Questions Quickly

Here i write some points on fast solve the Polynomail Questions.

How to Download Class 10 Maths Chapter 2 Polynomials Important Questions with Solutions (CBSE 2026 PDF)

i have provided the Download button bellow you can simply click and download pdf file. table of content also added. simple click from there or you can scrol and you find the button easly.

FAQs

• What is a polynomial?
• Ans-A polynomial is an algebraic expression made up of variables and coefficients, where the powers of variables are non-negative integers.
• What are the zeroes of a polynomial?
• Ans-zeroes are the values of $x$x for which the polynomial becomes zero, i.e., $p(x) = 0$.
• How can we find the number of zeroes from a graph?
• Ans-The number of times the graph intersects the x-axis gives the number of zeroes.
• What is the degree of a polynomial?
• Ans-The degree is the highest power of the variable in the polynomial.
• Can a polynomial have fractional or negative powers?
• Ans-No. A polynomial can only have non-negative integer powers of the variable.
• What is the Remainder Theorem?
• Ans-When a polynomial $p(x)$ is divided by $(x-a)$, the remainder is equal to $p(a)$.
• . What is the Factor Theorem?
• Ans-If $p(a) = 0$, then $(x-a)$ is a factor of the polynomial.
• Can a quadratic polynomial have no real zeroes?
• Ans-Yes. If the graph does not intersect the x-axis, it has no real zeroes.
• What is a constant polynomial?
• Ans-A polynomial with only a constant term (like 5) is called a constant polynomial. Its degree is 0.

Conclusion

Polynomials is one of the most important chapters in Class 10 Maths and forms the foundation for higher mathematics. From basic concepts like zeroes and degree to important theorems like Remainder Theorem and Factor Theorem, this chapter is highly scoring if prepared strategically.

Practice a variety of questions including HOTS, case-based, and previous year questions to build confidence.

Final Exam Preparation Tips

• Revise all key formulas and relationships regularly
• Practice previous year questions multiple times
• Focus on graph-based questions (very important for boards)
• Use Remainder & Factor Theorem to save time in exams
• Avoid calculation mistakes by checking signs carefully
• Attempt easy questions first to build momentum

With consistent practice and the right strategy, scoring high in Polynomials is absolutely achievable. Stay confident, practice smartly, and you can easily boost your overall Maths score.

thanks