- A variable represents a number; an algebraic expression combines numbers, variables and operations.
- A polynomial has nonnegative integer exponents of the variable and real coefficients.
- A linear polynomial has degree 1 and form p(x)=ax+b where a≠0.
- In ax+b, a is the coefficient of x and b is the constant term.
- Substitute a number for x and evaluate following order of operations.
- A zero of p(x) is a value r for which p(r)=0. For ax+b, r=−b/a.
- The graph y=ax+b is a nonvertical straight line; the zero corresponds to its x-intercept.
- The coefficient a is the slope and b is the y-intercept for y=ax+b.
- To solve ax+b=c, perform equivalent operations on both sides, with a≠0.
- Represent a fixed amount plus a rate per unit with a linear expression.
- Simplify like terms and test equivalence by algebra, not just one numerical value.
- The degree of a nonzero constant is zero; the zero polynomial's degree is undefined. Do not divide by a variable without checking whether it can be zero.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Introduction to Linear Polynomials — Class 9 Mathematics
Textbook edition: Ganita Manjari 2026–27, Part I, Chapter 2. This chapter is part of the new 2026–27 Ganita Manjari textbook.
Direct explanation: A variable represents a number; an algebraic expression combines numbers, variables and operations.
Original supplementary guide. The precise NCERT exercise numbers and every textbook activity have not been independently checked against a complete official PDF; the practice sets below are original, not falsely labelled textbook solutions.
Learning outcomes
- Understand definitions, notation and necessary conditions.
- Apply concepts in numerical and reasoning problems.
- Show each calculation and justify the method.
- Check answers and recognise common errors.
Variables and expressions
A variable represents a number; an algebraic expression combines numbers, variables and operations.
Worked example
3x+5 is an expression; x is the variable.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Polynomials in one variable
A polynomial has nonnegative integer exponents of the variable and real coefficients.
Worked example
2x²−3x+1 is a polynomial; 1/x is not a polynomial in x.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Linear polynomials
A linear polynomial has degree 1 and form p(x)=ax+b where a≠0.
Worked example
p(x)=5x−10 is linear.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Coefficients and constant term
In ax+b, a is the coefficient of x and b is the constant term.
Worked example
For −4x+7, coefficient −4 and constant 7.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Value at a given x
Substitute a number for x and evaluate following order of operations.
Worked example
If p(x)=3x−2, p(4)=10.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Zero or root
A zero of p(x) is a value r for which p(r)=0. For ax+b, r=−b/a.
Worked example
2x−6=0 gives x=3.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Graph of a linear polynomial
The graph y=ax+b is a nonvertical straight line; the zero corresponds to its x-intercept.
Worked example
For y=2x−4, the x-intercept is (2,0).
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Slope and intercept
The coefficient a is the slope and b is the y-intercept for y=ax+b.
Worked example
y=−3x+6 has slope −3 and y-intercept 6.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Solving linear equations
To solve ax+b=c, perform equivalent operations on both sides, with a≠0.
Worked example
3x+5=20 gives x=5.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Building expressions from contexts
Represent a fixed amount plus a rate per unit with a linear expression.
Worked example
A ₹30 fee plus ₹5 per kilometre is C=30+5k.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Comparing expressions
Simplify like terms and test equivalence by algebra, not just one numerical value.
Worked example
2(x+3)=2x+6 for every x.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Common mistakes
The degree of a nonzero constant is zero; the zero polynomial's degree is undefined. Do not divide by a variable without checking whether it can be zero.
Worked example
p(x)=0 for every x only for the zero polynomial.
Check your understanding
Explain the rule in your own words, then verify the worked example using an independent calculation or substitution.
Concept and formula reference
| Topic | Rule or definition | Example |
|---|---|---|
| Variables and expressions | A variable represents a number; an algebraic expression combines numbers, variables and operations. | 3x+5 is an expression; x is the variable. |
| Polynomials in one variable | A polynomial has nonnegative integer exponents of the variable and real coefficients. | 2x²−3x+1 is a polynomial; 1/x is not a polynomial in x. |
| Linear polynomials | A linear polynomial has degree 1 and form p(x)=ax+b where a≠0. | p(x)=5x−10 is linear. |
| Coefficients and constant term | In ax+b, a is the coefficient of x and b is the constant term. | For −4x+7, coefficient −4 and constant 7. |
| Value at a given x | Substitute a number for x and evaluate following order of operations. | If p(x)=3x−2, p(4)=10. |
| Zero or root | A zero of p(x) is a value r for which p(r)=0. For ax+b, r=−b/a. | 2x−6=0 gives x=3. |
| Graph of a linear polynomial | The graph y=ax+b is a nonvertical straight line; the zero corresponds to its x-intercept. | For y=2x−4, the x-intercept is (2,0). |
| Slope and intercept | The coefficient a is the slope and b is the y-intercept for y=ax+b. | y=−3x+6 has slope −3 and y-intercept 6. |
| Solving linear equations | To solve ax+b=c, perform equivalent operations on both sides, with a≠0. | 3x+5=20 gives x=5. |
| Building expressions from contexts | Represent a fixed amount plus a rate per unit with a linear expression. | A ₹30 fee plus ₹5 per kilometre is C=30+5k. |
| Comparing expressions | Simplify like terms and test equivalence by algebra, not just one numerical value. | 2(x+3)=2x+6 for every x. |
| Common mistakes | The degree of a nonzero constant is zero; the zero polynomial's degree is undefined. Do not divide by a variable without checking whether it can be zero. | p(x)=0 for every x only for the zero polynomial. |
Practice exercise A: Fundamental understanding
- State and explain variables and expressions. Give a new valid example and show why it satisfies the definition.
- State and explain polynomials in one variable. Give a new valid example and show why it satisfies the definition.
- State and explain linear polynomials. Give a new valid example and show why it satisfies the definition.
- State and explain coefficients and constant term. Give a new valid example and show why it satisfies the definition.
- State and explain value at a given x. Give a new valid example and show why it satisfies the definition.
- State and explain zero or root. Give a new valid example and show why it satisfies the definition.
- State and explain graph of a linear polynomial. Give a new valid example and show why it satisfies the definition.
- State and explain slope and intercept. Give a new valid example and show why it satisfies the definition.
- State and explain solving linear equations. Give a new valid example and show why it satisfies the definition.
- State and explain building expressions from contexts. Give a new valid example and show why it satisfies the definition.
Practice exercise B: Application and reasoning
- Study this example: 3x+5 is an expression; x is the variable. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: 2x²−3x+1 is a polynomial; 1/x is not a polynomial in x. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: p(x)=5x−10 is linear. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: For −4x+7, coefficient −4 and constant 7. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: If p(x)=3x−2, p(4)=10. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: 2x−6=0 gives x=3. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: For y=2x−4, the x-intercept is (2,0). Explain each step, then construct and solve a related problem with different valid values.
- Study this example: y=−3x+6 has slope −3 and y-intercept 6. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: 3x+5=20 gives x=5. Explain each step, then construct and solve a related problem with different valid values.
- Study this example: A ₹30 fee plus ₹5 per kilometre is C=30+5k. Explain each step, then construct and solve a related problem with different valid values.
Practice exercise C: Higher-order thinking
- Identify two concepts that are related and explain their differences with examples.
- Find an incorrect claim about the chapter and disprove it with a counterexample.
- Explain why a key formula or method works under its stated conditions.
- Design a real-world application and state its assumptions and units.
- Compare two ways of solving a suitable problem.
- Write a complete solution that a classmate can follow without extra explanation.
Frequently asked questions with model answers
Explain variables and expressions in Introduction to Linear Polynomials.
A variable represents a number; an algebraic expression combines numbers, variables and operations. Example: 3x+5 is an expression; x is the variable.
Explain polynomials in one variable in Introduction to Linear Polynomials.
A polynomial has nonnegative integer exponents of the variable and real coefficients. Example: 2x²−3x+1 is a polynomial; 1/x is not a polynomial in x.
Explain linear polynomials in Introduction to Linear Polynomials.
A linear polynomial has degree 1 and form p(x)=ax+b where a≠0. Example: p(x)=5x−10 is linear.
Explain coefficients and constant term in Introduction to Linear Polynomials.
In ax+b, a is the coefficient of x and b is the constant term. Example: For −4x+7, coefficient −4 and constant 7.
Explain value at a given x in Introduction to Linear Polynomials.
Substitute a number for x and evaluate following order of operations. Example: If p(x)=3x−2, p(4)=10.
Explain zero or root in Introduction to Linear Polynomials.
A zero of p(x) is a value r for which p(r)=0. For ax+b, r=−b/a. Example: 2x−6=0 gives x=3.
Explain graph of a linear polynomial in Introduction to Linear Polynomials.
The graph y=ax+b is a nonvertical straight line; the zero corresponds to its x-intercept. Example: For y=2x−4, the x-intercept is (2,0).
Explain slope and intercept in Introduction to Linear Polynomials.
The coefficient a is the slope and b is the y-intercept for y=ax+b. Example: y=−3x+6 has slope −3 and y-intercept 6.
Explain solving linear equations in Introduction to Linear Polynomials.
To solve ax+b=c, perform equivalent operations on both sides, with a≠0. Example: 3x+5=20 gives x=5.
Explain building expressions from contexts in Introduction to Linear Polynomials.
Represent a fixed amount plus a rate per unit with a linear expression. Example: A ₹30 fee plus ₹5 per kilometre is C=30+5k.
Explain comparing expressions in Introduction to Linear Polynomials.
Simplify like terms and test equivalence by algebra, not just one numerical value. Example: 2(x+3)=2x+6 for every x.
Explain common mistakes in Introduction to Linear Polynomials.
The degree of a nonzero constant is zero; the zero polynomial's degree is undefined. Do not divide by a variable without checking whether it can be zero. Example: p(x)=0 for every x only for the zero polynomial.
Revision checklist
- I can explain and apply variables and expressions.
- I can explain and apply polynomials in one variable.
- I can explain and apply linear polynomials.
- I can explain and apply coefficients and constant term.
- I can explain and apply value at a given x.
- I can explain and apply zero or root.
- I can explain and apply graph of a linear polynomial.
- I can explain and apply slope and intercept.
- I can explain and apply solving linear equations.
- I can explain and apply building expressions from contexts.
- I can explain and apply comparing expressions.
- I can explain and apply common mistakes.
Sources and accuracy
Official NCERT textbook portal. Original supplementary explanations and self-created practice exercises. Consult the prescribed edition for its actual numbered exercises, figures and activities; no official CBSE Class 9 board marking scheme is claimed.
Frequently asked questions
Explain variables and expressions in Introduction to Linear Polynomials.
A variable represents a number; an algebraic expression combines numbers, variables and operations. Example: 3x+5 is an expression; x is the variable.
Explain polynomials in one variable in Introduction to Linear Polynomials.
A polynomial has nonnegative integer exponents of the variable and real coefficients. Example: 2x²−3x+1 is a polynomial; 1/x is not a polynomial in x.
Explain linear polynomials in Introduction to Linear Polynomials.
A linear polynomial has degree 1 and form p(x)=ax+b where a≠0. Example: p(x)=5x−10 is linear.
Explain coefficients and constant term in Introduction to Linear Polynomials.
In ax+b, a is the coefficient of x and b is the constant term. Example: For −4x+7, coefficient −4 and constant 7.
Explain value at a given x in Introduction to Linear Polynomials.
Substitute a number for x and evaluate following order of operations. Example: If p(x)=3x−2, p(4)=10.
Explain zero or root in Introduction to Linear Polynomials.
A zero of p(x) is a value r for which p(r)=0. For ax+b, r=−b/a. Example: 2x−6=0 gives x=3.
Explain graph of a linear polynomial in Introduction to Linear Polynomials.
The graph y=ax+b is a nonvertical straight line; the zero corresponds to its x-intercept. Example: For y=2x−4, the x-intercept is (2,0).
Explain slope and intercept in Introduction to Linear Polynomials.
The coefficient a is the slope and b is the y-intercept for y=ax+b. Example: y=−3x+6 has slope −3 and y-intercept 6.
Explain solving linear equations in Introduction to Linear Polynomials.
To solve ax+b=c, perform equivalent operations on both sides, with a≠0. Example: 3x+5=20 gives x=5.
Explain building expressions from contexts in Introduction to Linear Polynomials.
Represent a fixed amount plus a rate per unit with a linear expression. Example: A ₹30 fee plus ₹5 per kilometre is C=30+5k.
Explain comparing expressions in Introduction to Linear Polynomials.
Simplify like terms and test equivalence by algebra, not just one numerical value. Example: 2(x+3)=2x+6 for every x.
Explain common mistakes in Introduction to Linear Polynomials.
The degree of a nonzero constant is zero; the zero polynomial's degree is undefined. Do not divide by a variable without checking whether it can be zero. Example: p(x)=0 for every x only for the zero polynomial.
