Introduction to Linear Polynomials Solutions for Class 9 Ganita Manjari

Introduction

Looking for easy and accurate Introduction to Linear Polynomials Solutions for Class 9 Ganita Manjari? This chapter helps students understand the basics of polynomials, degrees of polynomials, coefficients, constant terms, and linear expressions with solved examples and practice questions. These step-by-step solutions are useful for Bihar Board and CBSE Class 9 students preparing for school exams and competitive tests.

What are Linear Polynomials?

A linear polynomial is a polynomial of degree 1. It contains variables with the highest power equal to 1.

Examples of Linear Polynomials

• 2x + 5

• 4z − 3

• 5x − 3

Linear polynomials are one of the most important concepts in Class 9 Algebra and help students build a strong foundation in mathematics.

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Exercise Solutions – Introduction to Linear Polynomials

1. Find the Degree of the Following Polynomials

(i) 2x² − 5x + 3

Highest power of x = 2

Degree = 2

(ii) y³ + 2y − 1

Highest power of y = 3

Degree = 3

(iii) −9

This is a constant polynomial.

Degree = 0

(iv) 4z − 3

Highest power of z = 1

Degree = 1

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2. Write Polynomials of Degrees 1, 2, and 3

Polynomial of Degree 1

Example: 2x + 5

Polynomial of Degree 2

Example: x² + 3x + 1

Polynomial of Degree 3

Example: x³ − 2x² + x + 4

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3. Find the Coefficients of x² and x³

Given polynomial:

x⁴ − 3x³ + 6x² − 2x + 7

• Coefficient of x³ = −3

• Coefficient of x² = 6

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4. Find the Coefficient of z

Polynomial:

4z³ + 5z² − 11

There is no z-term.

Coefficient of z = 0

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5. Find the Constant Term

Polynomial:

9x³ + 5x² − 8x − 10

The constant term is the term without any variable.

Constant Term = −10

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Value of Linear and Quadratic Polynomials

1. Find the Value of 5x − 3

(i) x = 0

5(0) − 3 = −3

Answer = −3

(ii) x = −1

5(−1) − 3 = −8

Answer = −8

(iii) x = 2

5(2) − 3 = 7

Answer = 7

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2. Find the Value of 7s² − 4s + 6

(i) s = 0

Answer = 6

(ii) s = −3

Answer = 81

(iii) s = 4

Answer = 102

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Word Problems Based on Linear Polynomials

1. Salil and His Mother’s Ages

Let Salil’s age = x years

Mother’s age = 3x years

According to the question:

(x + 5) + (3x + 5) = 70

4x + 10 = 70

4x = 60

x = 15

Final Answer

• Salil’s age = 15 years

• Mother’s age = 45 years

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2. Difference Between Two Integers

Let the integers be 2x and 5x.

5x − 2x = 63

3x = 63

x = 21

Final Answer

• First integer = 42

• Second integer = 105

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3. Ruby’s Coins Problem

Let number of five-rupee coins = x

Number of two-rupee coins = 3x

5x + 2(3x) = 88

11x = 88

x = 8

Final Answer

• Five-rupee coins = 8

• Two-rupee coins = 24

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4. Farmer’s Fence Problem

Let shorter piece = x feet

Longer piece = 4x feet

x + 4x = 300

5x = 300

x = 60

Final Answer

• Shorter piece = 60 feet

• Longer piece = 240 feet

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5. Rectangle Dimensions Problem

Let width = x cm

Length = 2x + 3 cm

2[(2x + 3) + x] = 24

6x + 6 = 24

6x = 18

x = 3

Final Answer

• Width = 3 cm

• Length = 9 cm

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Linear Expressions and Patterns

1. Savings Bank Problem

Initial amount = ₹500

Pocket money every month = ₹150

Linear Expression

Aₙ = 150n + 500

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2. Rally Members Problem

Initial members = 120

Members leaving every hour = 9

Linear Expression

Mₙ = 120 − 9n

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3. Area of Rectangle Pattern

Length = 13 cm

Area = Length × Breadth

Linear Expression

A = 13x

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4. Volume of Rectangular Box Pattern

Length = 7 cm
Breadth = 11 cm

Volume = Length × Breadth × Height

Linear Expression

V = 77h

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Conclusion

These Introduction to Linear Polynomials Solutions for Class 9 Ganita Manjari help students understand polynomial concepts in a simple and easy way. By practicing these solved examples and word problems, students can improve their algebra skills and score better in exams. Regular revision of linear polynomials, coefficients, degrees, and linear expressions is important for mastering Class 9 Mathematics.

FAQs

What is a linear polynomial?

A linear polynomial is a polynomial whose highest power of the variable is 1.

What is the degree of a constant polynomial?

The degree of a constant polynomial is 0.

Why are linear polynomials important in Class 9 Maths?

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