CLASS 9 • MATHS • CHAPTER 2
Introduction to
Introduction to
Linear Polynomials
Short, colourful, handwritten-style notes — by EduGrown
1Algebraic Expressions — The Basics
Raju buys x red boxes (4 pens each) and y blue boxes (5 pencils each), and gets 3 extra pens free. Total pens + pencils = 4x + 5y + 3.
📝 Four words you must know
Look at the expression 4x + 5y + 3:
- Terms → the parts joined by + or – : 4x, 5y, 3
- Variables → the letters : x and y
- Coefficients → the numbers multiplying the variables : 4 and 5
- Constant → the plain number with no variable : 3
Another example — fencing a garden of length l and width w gives the total cost expression 200l + 160w + 50lw.
2Polynomials in One Variable & Degree
📝 Definition
An algebraic expression that uses only one variable and its powers is called a
one-variable polynomial or univariate polynomial — or simply a polynomial.
(‘Uni’ = one, ‘variate’ = variable.)
(‘Uni’ = one, ‘variate’ = variable.)
- 4x, x² + 1, 2y – 5, 3z + 7 → all are one-variable polynomials.
- 4x + 5y + 3 → two variables, so not studied in this chapter.
⭐ Degree
The highest power of the variable in a polynomial is called its degree.
Example: in 5y³ + y² + 2y – 1, the highest power is 3, so the degree is 3.
Example: in 5y³ + y² + 2y – 1, the highest power is 3, so the degree is 3.
✏️ Reading a polynomial
For 5y³ + y² + 2y – 1:
⇧ back to top
- Coefficient of y³ = 5
- Coefficient of y² = 1
- Coefficient of y = 2
- Constant term = –1
3Types of Polynomials by Degree
| Degree | Name | Example |
|---|---|---|
| 0 | Constant polynomial | 8 (can be written as 8x0) |
| 1 | Linear polynomial | 3z + 7 |
| 2 | Quadratic polynomial | x² + 5x + 1 |
| 3 | Cubic polynomial | 5y³ + y² + 2y – 1 |
💡 Memory trick
Constant → Linear → Quadratic → Cubic for degrees 0, 1, 2, 3. Just count up!
⇧ back to top
4Linear Polynomials
📝 Definition
A polynomial of degree 1 is called a linear polynomial.
General form: ax + b (where a ≠ 0).
General form: ax + b (where a ≠ 0).
- Perimeter of a square of side x = 4x → linear polynomial in x.
- A chess club charges ₹200 joining fee + ₹50 per match → total = 200 + 50m, a linear polynomial in m.
| Matches played (m) | 1 | 2 | 3 | 4 | 5 | … | m |
|---|---|---|---|---|---|---|---|
| Amount paid (₹) | 250 | 300 | 350 | 400 | 450 | … | 200 + 50m |
🔍 Key feature
Look at the amounts: 250, 300, 350, 400… They go up by the same ₹50 every time. In a linear polynomial the difference between successive values is always constant.
📝 Linear equation
When we equate a linear polynomial to a constant, we get a linear equation.
Example: 2x + 10 is a polynomial; 2x + 10 = 64 is a linear equation.
⇧ back to top
Example: 2x + 10 is a polynomial; 2x + 10 = 64 is a linear equation.
5Polynomial as an Input–Output Machine
Think of a polynomial as a machine: you put a value of x in, and it gives you an answer out.
✏️ Machine: 2x + 3
- Input x = 4 → 2 × 4 + 3 = 11
- Input x = –6 → 2 × (–6) + 3 = –9
📝 Function
Such an input–output process is called a function. Here 2x + 3 is a function of x.
2x + 3 is a linear function, while 10x – x² is a quadratic function.
⇧ back to top
2x + 3 is a linear function, while 10x – x² is a quadratic function.
6Linear Patterns
Look at this growing pattern of square tiles — each stage adds 2 more tiles:
| Stage (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Number of tiles | 1 | 3 | 5 | 7 | 9 | 11 | 13 |
⭐ The rule
Tiles at Stage n = 2n – 1
Check: Stage 5 → 2 × 5 – 1 = 9 ✔
Since 2n – 1 has degree 1, it is a linear polynomial.
Since 2n – 1 has degree 1, it is a linear polynomial.
📝 Linear pattern
A linear pattern is a sequence of numbers in which the difference between two consecutive terms is constant.
Here: 1, 3, 5, 7, 9… the difference is always 2.
Here: 1, 3, 5, 7, 9… the difference is always 2.
💡 More everyday linear patterns
⇧ back to top
- Bela has ₹100 and spends ₹5 daily → money left on day n = 100 – 5n
- Auto fare: ₹25 for the first 2 km, then ₹15 per km → fare for n km = 15n – 5 (for n ≥ 2)
7Linear Growth & Linear Decay
| Linear Growth 📈 | Linear Decay 📉 | |
|---|---|---|
| Meaning | Quantity increases by a fixed amount over equal intervals | Quantity decreases by a fixed amount over equal intervals |
| Example | Cost C(d) = 100 + 60d — cost rises by ₹60 per km | Water height h(t) = 3 – 0.5t — height falls by 0.5 m per month |
| On a graph | Straight line with positive slope (goes up ↗) | Straight line with negative slope (goes down ↘) |
8Linear Relationships: y = ax + b
⭐ The master form
A relationship between two variables x and y is linear if it can be written as
y = ax + b
where a = slope and b = y-intercept.
For the tile pattern, taking x = stage number and y = number of tiles, the relationship is y = 2x – 1.
✔ Finding a and b from two given facts
✏️ Worked example
A telecom company: 10 GB → bill ₹350; 20 GB → bill ₹550. Find y = ax + b.
⇧ back to top
- Substitute both facts: 350 = 10a + b and 550 = 20a + b
- From the first: b = 350 – 10a
- Put it in the second: 550 = 20a + (350 – 10a) → 10a = 200 → a = 20
- So b = 350 – 200 = 150
- Answer: y = 20x + 150 (₹20 per GB + ₹150 fixed fee)
9Plotting a Linear Equation
💡 Only 2 points needed!
A straight line is fixed by any two points. So pick two easy values of x, find y, plot, and join with a ruler — then extend the line both ways.
✏️ Plotting y = 2x + 1
- x = 0 → y = 1 → point A (0, 1)
- x = 3 → y = 7 → point B (3, 7)
⭐ Golden rule
If a point lies on a line, its coordinates must satisfy the equation of that line.
Check (7, 15) on y = 2x + 1 → 2(7) + 1 = 15 ✔ So the point lies on the line.
⇧ back to top
Check (7, 15) on y = 2x + 1 → 2(7) + 1 = 15 ✔ So the point lies on the line.
10Slope ‘a’ — What It Does
Every line of the form y = ax passes through the origin (0, 0). The number a is called the slope — it decides how steep the line is.
✔ When a is positive (a > 0)
- a > 1 → the line is steeper than y = x
- a < 1 → the line is less steep than y = x
- y = x is equally inclined to both axes (45°)
✔ When a is negative (a < 0)
⭐ Slope tells the story
⇧ back to top
- Positive slope → line rises ↗ → linear growth
- Negative slope → line falls ↘ → linear decay
- The slope also equals the constant difference between consecutive terms of the pattern. (For y = 2x – 1, slope = 2 — and the tiles went up by 2 each stage.)
11y-Intercept ‘b’ & Parallel Lines
📝 y-intercept
Any line y = ax + b cuts the y-axis at the point (0, b). This value b is called the y-intercept.
| Line | Cuts the y-axis at | y-intercept |
|---|---|---|
| y = 2x + 5 | A (0, 5) | 5 |
| y = x + 3 | B (0, 3) | 3 |
| y = 3x – 2 | C (0, –2) | –2 |
✔ Keep a the same, change b → parallel lines
⭐ Three conclusions
⇧ back to top
- In y = ax + b, a gives the slope and b gives the y-intercept.
- Change a, keep b fixed → the steepness changes, but the line still cuts the y-axis at the same point.
- Change b, keep a fixed → the line shifts up or down but stays parallel. So lines with equal slopes but different y-intercepts are parallel.
12Common Mistakes to Avoid
❌ Don’t do this
⇧ back to top
- Calling the power the coefficient. In 5y³, the coefficient is 5 and the degree is 3.
- Forgetting that a constant like 8 is a polynomial of degree 0, not degree 1.
- Forgetting the sign of the constant term. In 9x³ + 5x² – 8x – 10 the constant term is –10, not 10.
- Mixing up a and b in y = ax + b — a is the slope, b is the y-intercept.
- Plotting only one point. A straight line always needs at least two points.
- Thinking a “linear” expression can have x² in it. If any power is 2 or more, it is not linear.
13Chapter Summary — Quick Revision
🏆 All of Chapter 2 in 10 lines
- An algebraic expression combines numbers, variables and operation signs; its parts are terms, with coefficients and constants.
- Univariate polynomials use only one variable and its powers.
- The degree is the highest power of the variable.
- Degree 0 = constant, 1 = linear, 2 = quadratic, 3 = cubic.
- A linear polynomial has degree one, e.g. 2x + 3 and 5 – 4y.
- A linear pattern is a sequence where the difference between consecutive terms is constant.
- Linear growth = increases by a fixed amount; linear decay = decreases by a fixed amount, over equal intervals.
- A linear relationship between x and y is the straight line y = ax + b.
- a = slope; b = y-intercept (the line cuts the y-axis at (0, b)). If b = 0 the line passes through the origin.
- Growth → positive slope; decay → negative slope. Same a, different b → parallel lines.
✨ Happy Learning — @edugrown ✨