NCERT Class 9 Maths Ganita Manjari Chapter 2 Solutions – Introduction to Linear Polynomials

The NCERT Class 9 Maths Ganita Manjari textbook is designed to make mathematics simple, logical, and practical for students. One of the most important early chapters is “Introduction to Linear Polynomials”, which builds a strong foundation in algebra. Understanding this chapter is essential because it prepares students for higher-level topics like equations, graphs, and coordinate geometry.

In this blog, you will find detailed explanations, solved examples, important concepts, and exam-oriented preparation tips for mastering this chapter.

What is a Polynomial?

A polynomial is an algebraic expression made up of variables, constants, and powers combined using addition, subtraction, or multiplication.

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Examples:

• 2x+32x + 32x+3
• x2+5x+1x^2 + 5x + 1x2+5x+1
• 7y−47y - 47y−4

What are Linear Polynomials?

A linear polynomial is a polynomial of degree 1. This means the highest power of the variable is 1.

General Form:

ax+bax + bax+b

Where:

• a ≠ 0
• a and b are constants
• x is a variable

Examples:

• 2x+32x + 32x+3
• −5x+7-5x + 7−5x+7
• x−9x - 9x−9

Key Concepts in This Chapter

1. Degree of a Polynomial

The degree is the highest power of the variable.

• 2x+32x + 32x+3 → Degree = 1 (Linear)
• x2+1x^2 + 1x2+1 → Degree = 2 (Quadratic)

2. Value of a Polynomial

To find the value, substitute a number for the variable.

Example:
Find value of 2x+32x + 32x+3 when x=2x = 2x=2

Solution:
2(2)+3=72(2) + 3 = 72(2)+3=7

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3. Zero of a Polynomial

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

Example:

Find zero of 2x+42x + 42x+4

Solution:
2x+4=02x + 4 = 02x+4=0
2x=−42x = -42x=−4
x=−2x = -2x=−2

NCERT Class 9 Chapter Solutions (Sample Questions)

Question 1: Find value of polynomial

Find value of 4x−14x - 14x−1 when x=3x = 3x=3

Solution:  4(3)−1=114(3) - 1 = 114(3)−1=11

Question 2: Find zero of polynomial

Find zero of x−5x - 5x−5

Solution:
x−5=0x - 5 = 0x−5=0
x=5x = 5x=5

Question 3: Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’.
(i)   y = 4x, y = 2x, y = x 
(ii)  y = – 6x, y = – 3x, y = – x 
(iii) y = 5x, y = –5x 
(iv)  y = 3x – 1, y = 3x, y = 3x + 1
(v)   y = –2x – 3, y = –2x, y = 2x + 3 

Solution:
 (i) y = 4x, y = 2x, y = x 

Observation:
1. All lines pass through because b = 0.
2. Larger a → steeper line.

(ii)    y = – 6x, y = – 3x, y = – x 

Observation:
1. All lines pass through the origin.
2. Negative a → lines slope downward.
3. Larger magnitude of a → steeper downward slope.

(iii) y = 5x, y = –5x 

(iv) y = 3x – 1, y = 3x, y = 3x + 1

Observation:
1. Both lines pass through the origin.
2. Same steepness, opposite direction (mirror images).

(v) y = –2x – 3, y = –2x, y = 2x + 3 

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Observation:
1. First two lines are parallel (same slope 2).
2. Third line has a different slope → different direction.
3. b controls vertical position.

Why This Chapter is Important?

This chapter is the foundation of algebra and helps students in:

• Solving equations
• Understanding graphs
• Preparing for coordinate geometry
• Building logical thinking
• Cracking competitive exams like Olympiads and JEE (basics)

Exam Preparation Tips

Practice Daily

Solve at least 10 polynomial questions daily.

Focus on Concepts

Understand degree, zeros, and graphs clearly.

Solve NCERT Exercises

Do all textbook examples and back exercises.

Revise Formulas

Remember standard form and methods.

Common Mistakes to Avoid

• Confusing linear and quadratic polynomials
• Incorrect substitution of values
• Calculation mistakes
• Not checking final answers

Real-Life Applications

Linear polynomials are used in:

• Business profit calculations
• Speed and distance problems
• Budget planning
• Data analysis
• Computer programming

Benefits of Using These Solutions

• Easy understanding of tough concepts
• Step-by-step explanations
• Helpful for homework and exams
• Saves time in revision
• Improves confidence

Conclusion

NCERT Class 9 Maths Ganita Manjari Chapter 2 Solutions – Introduction to Linear Polynomials is a crucial chapter for building a strong algebra base. By practicing regularly and understanding key concepts like degree, zeros, and graphs, students can easily score high marks.

Make sure to revise regularly and solve all NCERT questions for the best results.

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