📚 Chapter at a Glance
📖 Introduction to Polynomials
In mathematics, we frequently come across expressions such as:
These are examples of algebraic expressions involving variables and numbers.
A special type of algebraic expression is called a polynomial.
🔢 Parts of a Polynomial
Consider the polynomial:
It contains different parts that are important to understand.
| Part | Meaning | Example |
|---|---|---|
| Variable | A quantity whose value may change | x |
| Coefficient | Number multiplying the variable | 5 |
| Constant | Number without a variable | 7 |
| Term | Part separated by + or − signs | 5x, 7 |
🧩 Understanding Terms
Consider:
The polynomial has two terms:
- 4x
- 9
Therefore, it is a binomial.
📊 Polynomials Based on Number of Terms
| Number of Terms | Name | Example |
|---|---|---|
| 1 | Monomial | 7x |
| 2 | Binomial | 3x + 4 |
| 3 | Trinomial | x² + 3x + 2 |
📐 Degree of a Polynomial
The degree of a non-zero polynomial is the highest power of the variable with a non-zero coefficient.
The highest power of x is 3.
Therefore:
📏 What is a Linear Polynomial?
A polynomial whose degree is exactly 1 is called a linear polynomial.
where:
- a and b are constants.
- a ≠ 0.
- The variable is usually represented by x.
✏️ Examples of Linear Polynomials
Highest power of x = 1
Therefore, it is a linear polynomial.
Highest power of x = 1.
Therefore, it is a linear polynomial.
Highest power of x = 1.
Therefore, it is a linear polynomial.
This can be written as:
Therefore, x is a linear polynomial.
🚫 Examples That Are Not Linear Polynomials
Degree = 2
Therefore, it is not linear.
Degree = 3
Therefore, it is not linear.
🔵 Constant Polynomial
A non-zero polynomial containing only a constant has degree 0.
The polynomial 7 has no variable term.
⭕ Zero Polynomial
A polynomial whose every coefficient is zero is called the zero polynomial.
🔢 Finding the Value of a Polynomial
We can find the value of a polynomial by substituting a particular value for the variable.
Find the value of:
when x = 4.
Substitute x = 4:
🎯 Zero of a Polynomial
A number is called a zero of a polynomial if the value of the polynomial becomes zero when that number is substituted for the variable.
Then a is called a zero of p(x).
🧮 Finding the Zero of a Linear Polynomial
Find the zero of:
For the zero:
Therefore:
Therefore, the zero is:
📌 General Form of a Linear Polynomial
where a and b are real numbers and a ≠ 0.
To find its zero:
🧠 More Solved Examples
Find the degree of:
Highest power of x = 2.
Determine whether:
is a linear polynomial.
The highest power of x is 1.
Find the zero of:
Set the polynomial equal to zero:
Find p(2), where:
📊 Quick Revision Table
| Concept | Important Point |
|---|---|
| Polynomial | Algebraic expression with suitable non-negative integer powers |
| Term | Part of a polynomial separated by + or − |
| Coefficient | Number multiplying the variable |
| Constant | Term without a variable |
| Degree | Highest power of the variable |
| Linear Polynomial | Polynomial of degree 1 |
| General Form | ax + b |
| Zero | Value of x for which polynomial becomes zero |
📝 Chapter Exercise – Introduction to Linear Polynomials
Test your understanding of the chapter with the following practice exercise.
(a) 5x + 3
(b) x² + 2
(c) 9x - 4
(d) x³ + 1
(a) x² + 2x + 1
(b) 7x + 5
(c) x³ - 4
(d) x² - 9
(a) Monomial
(b) Binomial
(c) Trinomial
(d) Linear polynomial
✅ Exercise Solutions
Polynomial:
Terms = 7x and 5
Coefficient of x = 7
Constant = 5
Polynomial:
The highest power of x is 3.
(a) 5x + 3 → Linear
(b) x² + 2 → Not linear, degree is 2.
(c) 9x - 4 → Linear
(d) x³ + 1 → Not linear, degree is 3.
The general form of a linear polynomial is:
where a ≠ 0.
Given:
For x = 2:
Answer: 11
Given:
Zero = 4
Given:
Zero = -3
Given:
For x = 5:
Answer = 3
The correct answer is:
Its highest power is 1, so it is a linear polynomial.
Answer: (b)
For a polynomial ax + b to be linear, the highest power of x must be 1.
If a = 0, then the expression becomes only b, which is a constant polynomial of degree 0.
Therefore:
The polynomial is:
The highest power of x is 1.
Given:
Zero = 3
Given:
For x = 3:
Answer = 20
Given:
The highest power of x is 2.
Therefore its degree is 2.
It is not a linear polynomial.
Examples:
(a) Monomial → 5x
(b) Binomial → 3x + 2
(c) Trinomial → x² + 3x + 2
(d) Linear polynomial → 4x - 7
🎯 Class 9 Maths Chapter 2 MCQs
Answer: B. 1
Answer: C. 3x + 5
Answer: B. 1
Answer: B. 8
Answer: C. 11
Answer: C. x² + 2x + 1
Answer: B. ax + b
Answer: B. -3
Answer: C. x² + 2
Answer: C. 4
Answer: C. 9
Answer: C. 8
🧠 Competency-Based Questions
A shopkeeper calculates the cost of an item using the expression:
Identify whether this is a linear polynomial.
Answer: Yes. Its degree is 1.
A student's score is represented by:
Find the value when x = 5.
A linear polynomial is:
At what value of x does it become zero?
📝 Class 9 Maths Chapter 2 Worksheet
Section A – Fill in the Blanks
1. A polynomial of degree 1 is called a __________.
2. The highest power of the variable gives the __________ of a polynomial.
3. The general form of a linear polynomial is __________.
4. In 7x + 4, the coefficient of x is __________.
5. In 8x + 5, the constant term is __________.
Section B – Identify the Degree
6. 4x + 5
7. 2x² + 3x + 1
8. 7x³ + x
Section C – Identify Linear Polynomials
9. 5x + 7
10. x² + 4
11. 9x - 2
12. x³ + 5
Section D – Find the Value
13. If p(x) = 2x + 5, find p(3).
14. If p(x) = 5x - 2, find p(4).
Section E – Find the Zero
15. Find the zero of 2x - 8.
16. Find the zero of 3x + 12.
17. Find the zero of 7x - 35.
Section F – Higher Order Thinking
18. Explain why x² + 3x + 2 is not a linear polynomial.
19. Can 5 be called a linear polynomial? Give a reason.
20. Find the value of x for which 4x + 16 becomes zero.
✅ Worksheet Answer Key
- Linear polynomial
- Degree
- ax + b
- 7
- 5
- 1
- 2
- 3
- Linear
- Not linear
- Linear
- Not linear
- 11
- 18
- 4
- -4
- 5
- Its degree is 2, not 1.
- No. It is a constant polynomial of degree 0.
- -4
⭐ Important Questions for Revision
- Define a polynomial.
- What is meant by the degree of a polynomial?
- Define a linear polynomial.
- Write the general form of a linear polynomial.
- What is a coefficient?
- What is a constant term?
- What is a zero of a polynomial?
- Find the zero of a given linear polynomial.
- Find the value of a polynomial for a given value of x.
- Differentiate between monomial, binomial and trinomial.
- Identify whether a polynomial is linear or not.
⚡ One-Minute Revision
Degree = 1
ax + b
a ≠ 0
p(x) = 0
-b/a
Degree = 0
Two terms
Three terms
🎯 Exam Tips
- First identify the highest power of the variable.
- If the highest power is 1, the polynomial is linear.
- Do not confuse the number of terms with the degree.
- For finding a zero, put the polynomial equal to zero.
- For finding p(a), substitute x = a carefully.
- Remember that in ax + b, a must not be zero for the polynomial to be linear.
- Always simplify your final answer.
❓ Frequently Asked Questions
What is Chapter 2 of Class 9 Maths?
The chapter is titled Introduction to Linear Polynomials.
What is a linear polynomial?
A polynomial of degree 1 is called a linear polynomial. Its general form is ax + b, where a ≠ 0.
What is the degree of 5x + 2?
The degree is 1 because the highest power of x is 1.
How do we find the zero of a polynomial?
Set the polynomial equal to zero and solve the resulting equation.
What is the zero of ax + b?
The zero of ax + b is -b/a, provided a ≠ 0.
Is x a linear polynomial?
Yes. x can be written as 1x + 0, so its degree is 1.
Is a constant polynomial linear?
No. A non-zero constant polynomial has degree 0, whereas a linear polynomial has degree 1.
📚 Class 9 Maths Chapter 2 Study Material
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This page includes chapter explanations, important definitions, solved examples, linear polynomial concepts, degree, coefficients, constants, zeros, chapter exercise questions with solutions, MCQs, competency-based questions and a practice worksheet.