CLASS 9 MATHS · CHAPTER 2 · STUDY NOTES
Introduction to Linear Polynomials
Understand introduction to linear polynomials through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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Use a letter for a quantity that can change
Build an expression from a simple situation.Packs in, cards out
Output y (units)
Input x (units)
- 6x + 2 · y = 6x + 2
Only whole non-negative x-values make sense for sealed packs.
Read the plotted values
| Input x | Output y |
|---|---|
| 0 | 2 |
| 1 | 8 |
| 2 | 14 |
| 3 | 20 |
| 4 | 26 |
- Variable
- A symbol for a quantity whose value may change.
- Coefficient
- The number multiplying a variable term.
If a pack contains 6 cards, x packs contain 6x cards. With 2 loose cards, the total is 6x + 2. Here 6x and 2 are terms; 6 is a coefficient; 2 is constant.
An expression gives a quantity. An equation says two expressions are equal. At x = 3, the expression gives 6 × 3 + 2 = 20 cards.
A different cost model
C = 30 + 8nA hypothetical craft kit has a fixed cost of ₹30 plus ₹8 per extra sheet. The constant is the fixed part; 8 is the per-sheet rate.
Take a moment. What do you think?
PAUSE & TRY · think first, then check with a free account
Question
What is the coefficient of x in 7 − 4x?
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The powers tell you whether it is a polynomial
Recognise a one-variable polynomial and find its degree.Terms sorted by power
- Polynomial
- A finite sum of terms axⁿ where n is a non-negative integer.
- Degree
- The highest power with a non-zero coefficient after simplifying.
| Expression | Degree / reason |
|---|---|
| 4x + 1 | 1: linear |
| 2x² + x | 2: quadratic |
| −7 | 0: non-zero constant |
| 1/x | Not a polynomial in x |
3x² − 2x + 5 has degree 2. A non-zero constant has degree 0. The zero polynomial has no degree in this convention.
1/x has power −1, and √x has power 1/2, so neither is a polynomial in x. First combine like terms: x² − x² + 4x has degree 1, not 2.
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Question
What is the coefficient of x in 2x³ + 5?
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A fixed increase makes a linear pattern
Connect a growing pattern with a rule.Each new stage adds 3
Objects (count)
Stage (number)
- 3n + 1 · Discrete whole-number stages.
Dots show stages. There is no stage 2.5 in this counting model.
Read the plotted values
| Stage | Objects |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
| 4 | 13 |
| 5 | 16 |
| 6 | 19 |
- Linear polynomial
- ax + b with a ≠ 0: degree 1.
- Constant difference
- The same change in output for equal changes in input.
The pattern 4, 7, 10, 13 grows by 3 each time. At stage n, the total is 4 + 3(n − 1) = 3n + 1. The first stage is n = 1, not zero.
Check the rule on early stages, then use it for a later one. A pattern with changing differences, such as 1, 4, 9, 16, is not linear.
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Question
How many objects are in stage 6 of 3n + 1?
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Put a value in and follow the operations
Evaluate a polynomial without guessing the output.Read each input as a point
Output y (units)
Input x (units)
- f(x) · y = 3x + 2
This abstract function allows negative as well as positive inputs.
Read the plotted values
| Input x | Output y |
|---|---|
| -2 | -4 |
| 4 | 14 |
- Function
- A rule giving one output for each allowed input.
For f(x) = 3x + 2, multiply the input by 3 and then add 2. f(−2) = −6 + 2 = −4. Brackets help keep a negative input clear.
The same idea works for a quadratic: g(x) = x² + 1 gives g(−2) = 5. Square the whole negative input; (−2)² is 4.
- 1
Write f(−2) = 3(−2) + 2.
- 2
Do the multiplication: −6 + 2.
- 3
Add to get −4.
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Question
What is f(0) for f(x) = 3x + 2?
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An equation asks which input works
Undo operations in the reverse order.Find the input for output 20
Output y (units)
Input x (units)
- 3x + 2 · y = 3x + 2
- Target 20 · y = 0x + 20
The intersection gives the solution x = 6.
Read the plotted values
| Input x | Output y |
|---|---|
| 0 | 2 |
| 8 | 26 |
| Input x | Output y |
|---|---|
| 0 | 20 |
| 8 | 20 |
- Solution
- A value that makes both sides of an equation equal.
For 3x + 2 = 20, subtract 2 from both sides: 3x = 18. Divide both sides by 3: x = 6.
Check by substitution: 3 × 6 + 2 really is 20. Keeping both sides equal is the reason each step works.
Two related numbers
x + (x + 8) = 44 → x = 18Let the smaller number be x. The larger is x + 8. Their sum gives 2x + 8 = 44, so the two numbers are 18 and 26.
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Question
Why should we subtract 2 from both sides, rather than one side?
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