Polynomials is one of the most important chapters in Class 10 Maths because it builds a strong base for algebra and helps students understand expressions, zeros, graphs, and relationships between coefficients and roots in a simple but powerful way. Many students think this chapter is easy in theory but still lose marks in exams because they get confused in graphical questions, forget the formula linking zeros and coefficients, or make mistakes while forming a polynomial from given zeros.
The good part is that Polynomials is one of the most scoring chapters in Class 10 when the basics are clear. Once students understand what a polynomial actually is, how it is classified, what zeros mean, and how the sum and product of zeros connect to the coefficients, most questions become direct and manageable.
At Deeksha Vedantu, we always encourage students to learn Polynomials in a structured way. First understand the language of the chapter, then practise zeros, then understand graphical meaning, and finally master the relationship between zeros and coefficients. With that sequence, the chapter becomes much easier.
Why Polynomials Is Important in Class 10
Polynomials is a board-relevant chapter and also a bridge chapter between basic algebra and quadratic equations.
Why Students Should Prepare This Chapter Well
| Reason | Why it matters |
| Board relevance | This chapter regularly appears in CBSE-style papers |
| Strong algebra base | It improves expression handling and factorisation confidence |
| Graph-based learning | Students connect algebra with visual understanding |
| Formula-based questions | Sum and product of zeros are frequently tested |
| Link with quadratic equations | This chapter directly supports later algebra topics |
Chapter Weightage and Scope
This chapter is part of algebra and becomes even more important when studied together with quadratic equations.
Weightage Table
| Area | Approximate importance |
| Polynomials alone | Around 3 to 4 marks |
| Polynomials with related algebra support | Around 6 to 7 marks when combined with quadratic-style understanding |
Chapter Overview at a Glance
This chapter becomes easier when students first see the complete structure together.
Quick Concept Table
| Topic | Main idea |
| Polynomial | Algebraic expression made using constants and variables |
| Types of polynomials | Classified by degree or number of terms |
| Value of a polynomial | Obtained by substituting a value of x |
| Zero of a polynomial | Value of x that makes the polynomial equal to zero |
| Graphical meaning of zeros | Points where the graph cuts the x-axis |
| Relationship between zeros and coefficients | Sum and product formulas for quadratic polynomials |
| Forming a polynomial from zeros | Use the standard quadratic expression |
What Is a Polynomial
A polynomial is an algebraic expression made by combining constants and variables using operations such as addition, subtraction, and multiplication, where the powers of the variables are whole numbers.
Polynomial Identification Table
| Expression | Polynomial or not | Why |
| 2 | Yes | Constant polynomial |
| x | Yes | Variable with whole-number power |
| x² + 3x + 2 | Yes | Standard polynomial form |
| x^1/2 + 2 | No | Power is fractional |
| 1/x + 3 | No | Variable appears in denominator |
Most Important Rule for a Polynomial
| Rule | Meaning |
| Power of the variable should be a whole number | Fractions, negatives, and variable denominators are not allowed in standard polynomial form |
Types of Polynomials Based on Number of Terms
This is one of the easiest and most direct classification methods.
Classification by Number of Terms
| Type | Number of terms | Example |
| Monomial | 1 | 2, x, 5x² |
| Binomial | 2 | x + 2, 3x² – 1 |
| Trinomial | 3 | x² + 5x + 6 |
A quadratic polynomial like ax² + bx + c is also a trinomial because it has three terms.
Types of Polynomials Based on Degree
The degree of a polynomial is the highest power of the variable present in it.
Classification by Degree Table
| Type | Degree | Example |
| Constant polynomial | 0 | 2 |
| Linear polynomial | 1 | x + 1 |
| Quadratic polynomial | 2 | x² + 3x + 2 |
| Cubic polynomial | 3 | x³ – 3x² + 4x + 1 |
Important Note on Zero Polynomial
| Case | Meaning |
| Zero polynomial | Degree is not defined |
This is because 0 can be written with any power and still remains 0.
Value of a Polynomial
The value of a polynomial is found by substituting a given value of x into the expression.
Solved Example 1: Find the Value of a Polynomial
Find P(1) and P(0) if P(x) = 35x – 2.
Given
| Quantity | Value |
| P(x) | 35x – 2 |
Step 1: Find P(1)
P(1) = 35(1) – 2 = 35 – 2 = 33
Step 2: Find P(0)
P(0) = 35(0) – 2 = -2
Answer
| Expression | Value |
| P(1) | 33 |
| P(0) | -2 |
Zeros of a Polynomial
A zero of a polynomial is that value of x which makes the polynomial equal to zero.
Zero of a Polynomial Table
| Statement | Meaning |
| If x = a is a zero of p(x) | Then p(a) = 0 |
| If x = a makes the expression zero | Then a is a zero of the polynomial |
Solved Example 2: Find the Zero of a Linear Polynomial
Find the zero of the polynomial p(x) = 5x + 2.
Given
| Polynomial | p(x) = 5x + 2 |
Step 1
Set the polynomial equal to zero:
5x + 2 = 0
Step 2
Solve for x:
5x = -2
x = -2/5
Answer
The zero of the polynomial is -2/5.
Solved Example 3: Find the Zeros of a Quadratic Polynomial
Find the zeros of p(x) = x² + 5x – 6.
Given
| Polynomial | x² + 5x – 6 |
Step 1
Set the polynomial equal to zero:
x² + 5x – 6 = 0
Step 2
Factorise:
x² + 6x – x – 6 = 0
x(x + 6) – 1(x + 6) = 0
(x + 6)(x – 1) = 0
Step 3
Find the values of x:
x + 6 = 0 or x – 1 = 0
x = -6 or x = 1
Answer
The zeros of the polynomial are -6 and 1.
Important Rule: Number of Zeros and Degree
The number of zeros of a polynomial cannot exceed its degree.
Degree and Maximum Zeros Table
| Polynomial degree | Maximum number of zeros |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| Constant polynomial | No zero in general |
Graphical Meaning of Zeros
This is one of the most important visual concepts of the chapter.
The zeros of a polynomial are the x-coordinates of the points where its graph cuts the x-axis.
Graphical Meaning Table
| Graph type | Number of x-axis intersections | Number of zeros |
| Line cuts x-axis once | 1 | 1 |
| Parabola cuts x-axis twice | 2 | 2 |
| Parabola touches x-axis once | 1 | 1 |
| Graph does not cut x-axis | 0 | 0 |
For Linear Polynomial
| Shape | Zero behaviour |
| Straight line | Cuts x-axis once, so one zero |
For Quadratic Polynomial
| Shape | Zero behaviour |
| Parabola | Can have 0, 1, or 2 real zeros |
Solved Example 4: Graph-Based Zero Interpretation
A graph of y = p(x) cuts the x-axis at four points. Find the number of zeros.
Given
| Observation | Graph cuts the x-axis at 4 points |
Step 1
Zeros of a polynomial are the x-axis intersection points.
Answer
The polynomial has 4 zeros.
Relationship Between Zeros and Coefficients
This is one of the most important formula-based sections of the chapter.
For a quadratic polynomial:
ax² + bx + c
if the zeros are α and β, then:
Formula Table
| Quantity | Formula |
| Sum of zeros | α + β = -b/a |
| Product of zeros | αβ = c/a |
These two formulas are enough to solve many board questions from this chapter.
Solved Example 5: Find Sum and Product of Zeros
If α and β are the zeros of 5x² – 6x + 1, find α + β and αβ.
Given
| a | 5 |
| b | -6 |
| c | 1 |
Step 1: Find the Sum of Zeros
α + β = -b/a = -(-6)/5 = 6/5
Step 2: Find the Product of Zeros
αβ = c/a = 1/5
Answer
| Quantity | Value |
| α + β | 6/5 |
| αβ | 1/5 |
Forming a Quadratic Polynomial from Given Zeros
If the zeros are α and β, then the required quadratic polynomial is:
x² – (α + β)x + αβ
Standard Formation Table
| Given | Required polynomial |
| Zeros α and β | x² – (α + β)x + αβ |
Solved Example 6: Form the Polynomial from Given Zeros
Form a quadratic polynomial whose zeros are 2 and 3.
Given
| Zero 1 | 2 |
| Zero 2 | 3 |
Step 1
Find the sum:
2 + 3 = 5
Step 2
Find the product:
2 × 3 = 6
Step 3
Use the formula:
x² – (sum of zeros)x + product of zeros
x² – 5x + 6
Answer
The required quadratic polynomial is x² – 5x + 6.
Solved Example 7: Find Unknown Value Using a Zero
If -3 is a zero of the polynomial 5x² – 6x + k, find k.
Given
| Polynomial | 5x² – 6x + k |
| Zero | -3 |
Step 1
Use p(-3) = 0.
Step 2
Substitute x = -3:
5(-3)² – 6(-3) + k = 0
5(9) + 18 + k = 0
45 + 18 + k = 0
63 + k = 0
k = -63
Answer
The value of k is -63.
Solved Example 8: Find Two Unknowns from Two Given Zeros
If 2 and -3 are the zeros of x² + (a + 1)x + b, find a and b.
Given
| Zero 1 | 2 |
| Zero 2 | -3 |
| Polynomial | x² + (a + 1)x + b |
Step 1: Use Sum of Zeros
2 + (-3) = -1
Also, sum = -(a + 1)/1 = -(a + 1)
So:
-(a + 1) = -1
a + 1 = 1
a = 0
Step 2: Use Product of Zeros
2 × (-3) = -6
So:
b = -6
Answer
| Quantity | Value |
| a | 0 |
| b | -6 |
Assertion-Reason and MCQ Logic from This Chapter
This chapter often appears in objective formats because the concepts are highly direct.
Common MCQ Areas Table
| Area | What may be asked |
| Type of polynomial | Identify by degree or number of terms |
| Zero of a polynomial | Value of x that makes p(x) = 0 |
| Graphical meaning | Number of x-axis intersections |
| Sum and product formulas | Direct calculation |
| Polynomial formation | Use given zeros to build equation |
Common Mistakes Students Make in Polynomials
Common Mistakes Table
| Mistake | Correct idea |
| Treating fractional powers as polynomial terms | Power must be a whole number |
| Forgetting to set p(x) = 0 while finding zeros | Zero means the polynomial becomes zero |
| Mixing up sum and product formulas | Sum is -b/a, product is c/a |
| Wrong sign while forming polynomial from zeros | Use x² – (sum)x + product |
| Confusing graph intersections with y-axis points | Zeros come only from x-axis intersections |
| Forgetting that constant polynomial usually has no zero | A fixed constant cannot usually become zero |
Quick Revision Sheet
This section is useful for one-glance revision before tests and board exams.
Quick Revision Table
| Topic | Key idea or formula |
| Polynomial | Algebraic expression with whole-number powers |
| Monomial | One term |
| Binomial | Two terms |
| Trinomial | Three terms |
| Degree | Highest power of the variable |
| Zero of polynomial | Value of x that makes p(x) = 0 |
| Graphical meaning of zero | Point where graph cuts x-axis |
| Sum of zeros | α + β = -b/a |
| Product of zeros | αβ = c/a |
| Form polynomial from zeros | x² – (α + β)x + αβ |
Important Practice Questions
Practice Set Table
| Question type | Practice prompt |
| Classification | Identify whether x² + 1/x is a polynomial or not |
| Value of polynomial | Find p(2) if p(x) = x² – 5x + 6 |
| Zeros of polynomial | Find the zeros of x² – 7x + 12 |
| Graphical meaning | If a graph cuts x-axis at 3 points, how many zeros does it have? |
| Sum and product | Find the sum and product of zeros of 4x² + 3x + 7 |
| Forming polynomial | Form the quadratic polynomial whose zeros are -1 and 5 |
| Unknown coefficient | If 1 is a zero of x² + kx – 2, find k |
FAQs
Q1. What is a polynomial in Class 10 Maths?
A polynomial is an algebraic expression in which the powers of the variables are whole numbers.
Q2. What is the zero of a polynomial?
A zero of a polynomial is the value of x that makes the polynomial equal to zero.
Q3. How do I know whether an expression is a polynomial or not?
Check the powers of the variables. If the powers are whole numbers and the variable is not in the denominator, it is a polynomial.
Q4. What is the graphical meaning of zeros of a polynomial?
The zeros are the x-coordinates of the points where the graph cuts the x-axis.
Q5. What are the formulas for sum and product of zeros?
For ax² + bx + c, the formulas are:
α + β = -b/a and αβ = c/a
Q6. How do I form a quadratic polynomial from given zeros?
If the zeros are α and β, the polynomial is x² – (α + β)x + αβ.
Q7. Can a constant polynomial have a zero?
A non-zero constant polynomial generally has no zero because it never becomes zero for any value of x.
Q8. How can I score full marks in this chapter?
Revise the classification properly, practise zeros and graph-based questions, remember the formulas for sum and product of zeros, and solve formation-based questions carefully.
Conclusion
Polynomials is one of the most scoring and concept-friendly chapters in Class 10 Maths when students learn it in the right order. The chapter may look simple, but it includes important exam ideas such as classification, zeros, graphs, and coefficient relationships. Once these concepts are clear, even board-style questions become much easier.
The best way to prepare this chapter is to understand what makes an expression a polynomial, practise finding zeros, revise the graphical meaning of zeros carefully, and master the formulas linking zeros and coefficients. At Deeksha Vedantu, we always remind students that in Polynomials, strong basics create strong answers.
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