1. Chapter at a Glance
A taxi charges ₹50 to start and ₹15 for every kilometre. A matchstick pattern adds 3 sticks for each new square. A candle burns down 2 cm every hour. In each case one quantity changes at a constant rate as another grows — and every such relationship is described by a linear polynomial of the form ax + b. This chapter introduces polynomials in general and then focuses on linear ones: their parts, values, zeros, patterns, tables and straight-line graphs.
- Algebraic expressions, terms, coefficients and constants.
- What makes an expression a polynomial; degree of a polynomial.
- Classifying polynomials by degree and by number of terms.
- Linear polynomials ax + b: value, zero, and rate of change.
- Finding the rule of a growing pattern.
- Graph of y = ax + b — always a straight line.
2. Expressions and Polynomials
An algebraic expression combines numbers and variables using +, −, ×, ÷. In 4x² − 3x + 7:
- Terms: 4x², −3x, 7.
- Coefficients: 4 (of x²), −3 (of x).
- Constant term: 7.
An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the powers of x are whole numbers (0, 1, 2, …) and the coefficients are real numbers.
| Expression | Polynomial? | Reason |
|---|---|---|
| 3x² + 5x − 1 | Yes | Powers 2, 1, 0 |
| √2·x + 7 | Yes | Coefficient may be irrational; power is 1 |
| x + 1/x | No | 1/x = x⁻¹ (negative power) |
| √x + 4 | No | √x = x^(1/2) (fractional power) |
| 9 | Yes | Constant polynomial (9x⁰) |
3. Degree and Classification
The degree of a polynomial in one variable is the highest power of the variable with a non-zero coefficient.
| By degree | General form | Example |
|---|---|---|
| Constant (degree 0) | c (c ≠ 0) | −5 |
| Linear (degree 1) | ax + b, a ≠ 0 | 3x − 2 |
| Quadratic (degree 2) | ax² + bx + c, a ≠ 0 | x² − 4 |
| Cubic (degree 3) | ax³ + bx² + cx + d, a ≠ 0 | 2x³ + x |
| By number of terms | Example |
|---|---|
| Monomial (1 term) | 7x³ |
| Binomial (2 terms) | 5x − 9 |
| Trinomial (3 terms) | x² + 2x + 1 |
The zero polynomial (0) has no defined degree.
For polynomials in more than one variable, the degree of a term is the sum of the powers: 3x²y³ has degree 5.
4. Linear Polynomials
A linear polynomial is p(x) = ax + b with a ≠ 0. Here a tells how much p(x) changes when x increases by 1 (the rate of change), and b is the value when x = 0 (the starting value).
Fare = ₹50 + ₹15 per km. Write it as a polynomial and find the fare for 12 km.
Solution: F(x) = 15x + 50. F(12) = 180 + 50 = ₹230. Here a = 15 (₹ per km), b = 50 (starting charge).
Value of a polynomial
Substituting a number for x gives the value of the polynomial at that number. For p(x) = 4x − 6: p(2) = 2, p(0) = −6, p(−1) = −10.
Zero of a polynomial
A number k is a zero of p(x) if p(k) = 0.
The linear polynomial ax + b has exactly one zero: x = −b/a.
Find the zero of p(x) = 3x + 12, and check.
Solution: 3x + 12 = 0 → x = −4. Check: p(−4) = −12 + 12 = 0 ✓.
A constant polynomial (non-zero) has no zero. Every real number is a zero of the zero polynomial.
5. Patterns that Grow Linearly
When each step adds the same amount, the rule is linear. Example — matchstick squares in a row:
| Squares (n) | 1 | 2 | 3 | 4 | n |
|---|---|---|---|---|---|
| Sticks | 4 | 7 | 10 | 13 | 3n + 1 |
| Difference | — | 3 | 3 | 3 | constant |
- Check that the differences are constant — that constant is a.
- Find b by working back to n = 0 (or b = first value − a).
- Rule: an + b. Test it on another value.
6. Graph of a Linear Polynomial
Plot the points (x, ax + b). They always lie on a straight line — that is why such polynomials are called "linear".
- The graph crosses the y-axis at b (the y-intercept).
- It crosses the x-axis at the zero −b/a.
- a > 0: line rises left to right; a < 0: line falls. Bigger |a| → steeper line.
- Lines with the same a are parallel.
7. Solved Problems (Board Pattern)
A 30 cm candle burns 2.5 cm per hour. Write its height after t hours. When will it burn out?
Solution: h(t) = 30 − 2.5t. Zero: 30 − 2.5t = 0 → t = 12 hours.
A linear polynomial p(x) has p(0) = −4 and p(2) = 6. Find p(x) and its zero.
Solution: b = −4. 2a − 4 = 6 → a = 5. p(x) = 5x − 4; zero = 4/5.
Plan A: ₹200 + ₹5 per GB. Plan B: ₹100 + ₹10 per GB. For how many GB do they cost the same?
Solution: 200 + 5x = 100 + 10x → x = 20 GB (both ₹300). Beyond 20 GB, Plan A is cheaper.
8. Master Formula Table
| Idea | Rule |
|---|---|
| Linear polynomial | p(x) = ax + b, a ≠ 0 |
| Zero of ax + b | x = −b/a |
| Rate of change | a = p(x + 1) − p(x) |
| Starting value | b = p(0) |
| Rule from table | a = common difference, b = first term − a |
| Degree of term xᵐyⁿ | m + n |
| Graph of y = ax + b | Straight line; y-intercept b; x-intercept −b/a |
9. Master Summary Table
| Concept | Key fact | Exam focus |
|---|---|---|
| Polynomial | Whole-number powers only | Identify polynomials |
| Degree | Highest power | Classify: linear/quadratic/cubic |
| Value | Substitute x | p(−1), p(0), p(2) |
| Zero | p(k) = 0; ax + b → −b/a | Find and verify zeros |
| Patterns | Constant difference → linear | Find nth term |
| Graph | Straight line | Intercepts, rising/falling |
10. Mind Map
- Terms, coefficients
- Degree, types
- a = rate
- b = start
- x = −b/a
- Exactly one
- Straight line
- Intercepts
11. Tips and Tricks
- To test if an expression is a polynomial, rewrite roots and fractions as powers: if any power is negative or fractional, it is not.
- Two points are enough to draw a line; plot a third to check.
- In word problems, the "per unit" amount is a and the fixed amount is b.
- Saying the zero of 2x + 6 is 3 (it is −3).
- Calling √x + 1 a polynomial.
- Thinking the degree of 5 is 1 — a non-zero constant has degree 0.
12. Important One-liners
- A polynomial has only whole-number powers of the variable.
- The degree of a non-zero constant polynomial is 0.
- A linear polynomial has exactly one zero, −b/a.
- The graph of a linear polynomial is a straight line.
- Constant differences in a table signal a linear rule.
- p(0) gives the constant term of any polynomial.
13. Exam-pattern Questions with Detailed Answers
Which is a linear polynomial?
- x² + 3
- 5 − 2x
- 1/x + 2
- x³
Show answer
5 − 2x has degree 1.
Answer: (B)
Find the degree of 7x⁴ − 3x² + x − 9.
Show answer
Highest power is 4.
Answer: 4
Find the zero of p(x) = 5x − 15.
Show answer
5x = 15 → x = 3.
Answer: 3
If p(x) = 3x − 7, find p(2) + p(−2) − p(0).
Show answer
p(2) = −1, p(−2) = −13, p(0) = −7. Sum: −1 − 13 + 7 = −7.
Answer: −7
Find the rule for the sequence of values 5, 9, 13, 17, … in terms of n (n = 1, 2, 3, …). What is the 25th value?
Show answer
Difference 4 → a = 4; b = 5 − 4 = 1. Rule 4n + 1. 25th: 101.
Answer: 4n + 1; 101
Is x = −2/3 a zero of p(x) = 3x + 2? Verify.
Show answer
p(−2/3) = −2 + 2 = 0. Yes.
Answer: Yes
Write the degree of 4x³y² − 2xy⁴ + 9.
Show answer
Term degrees: 5, 5, 0. Highest = 5.
Answer: 5
Where does the graph of y = −3x + 6 cut the two axes? Does it rise or fall?
Show answer
y-intercept: (0, 6). x-intercept: −3x + 6 = 0 → (2, 0). Since a = −3 < 0, it falls.
Answer: (0, 6) and (2, 0); falls
A linear polynomial p(x) satisfies p(1) = 4 and p(4) = 13. Find p(x), its zero and p(10).
Show answer
a = (13 − 4)/(4 − 1) = 3; b = 4 − 3 = 1. p(x) = 3x + 1. Zero: −1/3. p(10) = 31.
Answer: p(x) = 3x + 1, zero −1/3, p(10) = 31
In a pattern of triangles in a row (sharing sides), 1 triangle needs 3 sticks, 2 need 5, 3 need 7. How many triangles can be made with 101 sticks?
Show answer
Rule: 2n + 1. 2n + 1 = 101 → n = 50.
Answer: 50 triangles
Temperature in °F is a linear polynomial in °C: water freezes at 0 °C = 32 °F and boils at 100 °C = 212 °F. Find the rule, and the temperature that has the same number in both scales.
Show answer
a = (212 − 32)/100 = 9/5, b = 32. F = (9/5)C + 32. Same value: C = (9/5)C + 32 → −(4/5)C = 32 → C = −40.
Answer: F = (9/5)C + 32; −40°
For what value of k is x = 2 a zero of p(x) = kx − 3k + 8? Also find p(5) for that k.
Show answer
p(2) = 2k − 3k + 8 = −k + 8 = 0 → k = 8. Then p(x) = 8x − 16, p(5) = 24.
Answer: k = 8; p(5) = 24
14. Quick Revision Summary
- Polynomials have whole-number powers; degree = highest power.
- Linear polynomial ax + b: a = rate of change, b = starting value.
- Exactly one zero, −b/a; verify by substituting.
- Constant differences → linear rule an + b.
- Graph is a straight line; y-intercept b, x-intercept −b/a.