Polynomials Class 10
Download the PDF containing Polynomials Class 10 notes and a mind map for chapter revision. Understand polynomial types, zeroes, graphs and coefficient relationships through the explanations, formulas and worked examples below.
What makes an expression a polynomial? What does its graph tell you about its zeroes? How can you form a quadratic polynomial without knowing its zeroes individually? Class 10 Polynomials connects these questions through algebra and graphs.
These Polynomials Class 10 notes explain the concepts needed to identify a polynomial, determine its degree and use the relationship between zeroes and coefficients. Read the notes for understanding, then use the Polynomials Class 10 mind map to recall the chapter’s main connections before practising questions.
What Is a Polynomial in Class 10?
A polynomial in one variable x is a finite algebraic expression whose variable powers are non-negative integers. Its coefficients can be real numbers. For example, 3x2 − 5x + 2 is a polynomial, but 1/x and √x are not polynomials in x because their powers of x are −1 and 1/2 respectively.
The degree of a non-zero polynomial is the highest power of the variable with a non-zero coefficient after simplifying. In 4x3 − 2x + 7, the degree is 3 and the constant term is 7. An irrational coefficient is allowed: √2x + 3 is a linear polynomial.
Types of Polynomials
| Type | Degree | Example |
|---|---|---|
| Non-zero constant | 0 | 7 |
| Linear | 1 | 2x − 5 |
| Quadratic | 2 | x2 − 4x + 3 |
| Cubic | 3 | x3 + 2x − 1 |
Polynomials can also be classified by the number of non-zero terms: a monomial has one, a binomial has two and a trinomial has three. These classifications differ from degree. For example, x5 + 1 is a binomial of degree 5.
Zeroes of a Polynomial and Their Geometrical Meaning
A number k is a zero of p(x) if p(k) = 0. On the graph of y = p(x), a real zero is the x-coordinate of a point where the graph meets the x-axis. The graph may cross the axis or touch it.
Example 1: Checking Whether a Number Is a Zero
Let p(x) = x2 − 5x + 6.
p(2) = 22 − 5 × 2 + 6
p(2) = 4 − 10 + 6
p(2) = 0
Therefore, 2 is a zero of p(x).
- A linear polynomial has exactly one real zero.
- A quadratic polynomial can have zero, one or two distinct real zeroes.
- A non-zero polynomial of degree n has at most n distinct real zeroes.
A Quadratic Does Not Always Have Two Distinct Real Zeroes
x2 − 1 has two distinct real zeroes: −1 and 1. x2 has one distinct real zero, 0, repeated twice. x2 + 1 has no real zeroes because its value is positive for every real x.
Class 10 Polynomials Formulas
If α and β are the zeroes of ax2 + bx + c, where a ≠ 0, use the following relationships. Alpha and beta are simply symbols representing the two zeroes, including repetition when appropriate.
| Quantity | Formula | Condition |
|---|---|---|
| Sum of zeroes | α + β = −b/a | a ≠ 0 |
| Product of zeroes | αβ = c/a | a ≠ 0 |
| Monic polynomial from sum S and product P | x2 − Sx + P | Leading coefficient is 1. |
| Sum of squares of zeroes | α2 + β2 = (α + β)2 − 2αβ | Useful without finding each zero separately. |
| Sum of reciprocals | 1/α + 1/β = (α + β)/(αβ) | Both zeroes must be non-zero. |
Class 10 Polynomials Questions with Solutions
Example 2: Find Zeroes and Verify the Relationships
Question: Find the zeroes of 2x2 − 7x + 3 and verify their sum and product.
2x2 − 7x + 3 = 0
2x2 − 6x − x + 3 = 0
2x(x − 3) − (x − 3) = 0
(2x − 1)(x − 3) = 0
x = 1/2 or x = 3
Verification:
a = 2, b = −7, c = 3
Sum = 1/2 + 3 = 7/2
−b/a = −(−7)/2 = 7/2
Product = (1/2) × 3 = 3/2
c/a = 3/2
Both relationships are verified.
Example 3: Form a Quadratic Polynomial
Question: Form a quadratic polynomial whose zeroes have sum 5 and product 6.
Required monic polynomial = x2 − Sx + P
Required polynomial = x2 − 5x + 6
Any non-zero constant multiple, such as 2x2 − 10x + 12, has the same zeroes. The monic form is a convenient choice unless another leading coefficient is specified.
Example 4: Find an Unknown Coefficient
Question: One zero of x2 + kx − 6 is 2. Find k and the other zero.
Since 2 is a zero, p(2) = 0.
4 + 2k − 6 = 0
2k − 2 = 0
k = 1
Product of zeroes = −6
2β = −6
β = −3
Example 5: Evaluate an Expression Using the Zeroes
Question: If α and β are the zeroes of x2 − 4x + 1, find α2 + β2.
α + β = 4
αβ = 1
α2 + β2 = (α + β)2 − 2αβ
α2 + β2 = 16 − 2
α2 + β2 = 14
Polynomials Class 10 Mind Map for Revision
Use the mind map of Polynomials Class 10 to connect definitions, degree, zeroes, graphs and coefficient relationships. Start with the meaning of a polynomial, recall how its degree is identified, and then link a zero to both p(k) = 0 and an x-axis intersection.
For active revision, cover the formula branch and write the sum and product formulas from memory. Then explain one example aloud. The mind map is most useful when you can expand each short point into a complete explanation or calculation.
Class 10 Polynomials Extra Questions and MCQs
-
Find the degree of 7x4 − 3x2 + 2.
Answer: 4. -
Find the zeroes of x2 − 9.
Answer: 3 and −3. -
Find the sum and product of the zeroes of
3x2 + 5x − 2.
Answer: Sum = −5/3; product = −2/3. -
Form a monic quadratic polynomial with zeroes 4 and −1.
Answer: x2 − 3x − 4.
Quick MCQ Check
1. Which expression is not a polynomial in x?
A. x2 + 1 B. √3x + 2
C. 1/x + 2 D. 5
Answer: C. It contains x with a negative exponent.
2. If a quadratic graph touches the x-axis at one point,
how many distinct real zeroes does it have?
A. 0 B. 1 C. 2 D. 3
Answer: B. The touching point represents a repeated zero.
NCERT Exercises and Test Preparation
Use these notes while practising Class 10 Polynomials Ex 2.1 and Ex 2.2 in your textbook. Graph-based questions require careful counting of x-axis intersections. Algebraic questions require correct coefficients, factorisation and verification of the zero–coefficient relationships.
For a Class 10 Polynomials test paper, revise definitions and signs before attempting calculations. Follow routine exercises with important questions involving unknown coefficients, polynomial formation and expressions in α and β. When practising previous year questions with solutions, check the stated board and year before treating them as verified PYQs.
Related Revision and Practice Resources
Explore the CBSE Class 10 worksheet collection and select Polynomials for further practice after revising these notes.
Revise number-system foundations with Real Numbers Class 10 Notes and Mind Map , or browse the Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this Polynomials Class 10 PDF contain notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for quick visual revision. Use the notes first, then recall the main concepts through the map before solving questions.
What is a polynomial in Class 10?
A polynomial in x is a finite expression with non-negative integer powers of x. For example, 2x2 − x + 5 is a polynomial. Expressions such as 1/x and √x are not polynomials in x.
What do alpha and beta mean in Polynomials?
α and β are names used for the two zeroes of a quadratic polynomial. They are not coefficients. If the polynomial is ax2 + bx + c, its zeroes satisfy α + β = −b/a and αβ = c/a.
How do I find the number of zeroes from a graph?
Count the distinct points where the graph meets the x-axis. A point where it touches the axis also counts. An intersection with the y-axis does not identify a zero unless that point is also on the x-axis.
Is zero a zero of every polynomial?
No. Zero is a zero of p(x) only when p(0) = 0. For x2 − 3x, this is true. For x2 − 3x + 2, p(0) = 2, so it is false.
What is the degree of the zero polynomial?
Its degree is undefined in the usual school convention because it has no term with a non-zero coefficient. A non-zero constant polynomial, such as 6, has degree 0.
Can a polynomial have irrational coefficients?
Yes. √2x + 3 is a polynomial because the power of x is 1. The restriction applies to variable exponents, not to whether the coefficients are rational or irrational.
Why is the sum of zeroes −b/a rather than b/a?
Expanding a(x − α)(x − β) gives ax2 − a(α + β)x + aαβ. Comparing the coefficient of x with b gives b = −a(α + β), so α + β = −b/a.
Can different quadratic polynomials have the same zeroes?
Yes. Multiplying a polynomial by a non-zero constant preserves its zeroes. For example, x2 − 5x + 6 and 3x2 − 15x + 18 both have zeroes 2 and 3.
How should I revise Polynomials Class 10 important questions?
Practise finding zeroes, checking coefficient relationships, reading graphs and forming polynomials. Then try unknown-coefficient questions and expressions such as α2 + β2. Explain the method before checking the final answer.
How can I avoid sign mistakes in Exercise 2.2?
Write the polynomial in descending powers and record a, b and c with their signs. For 2x2 − 7x + 3, b = −7, so −b/a = 7/2. Use brackets when substituting negative values.
Are the notes and mind map enough for exam preparation?
They support understanding and recall, but you should also solve textbook exercises and practice questions independently. Use a test paper to check whether you can apply the formulas without referring to the notes.
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