Introduction to Linear Polynomials · Lesson 2 of 7
Linear Polynomials
“A polynomial with just one degree—because not every expression needs drama.”
• Recognise and write linear polynomials in one variable. • Evaluate a linear polynomial for given input values. • Understand the constant-change behaviour of linear polynomials. • Connect linear polynomials with linear equations. • Interpret a linear polynomial as an input-output process.
A linear polynomial is a polynomial of degree 1. Its variable appears only to the first power. The simplest general form is ax + b, where a is non-zero. The coefficient a controls how the value changes when x changes, while b contributes a fixed amount.
A polynomial of degree 1, generally written as ax + b with a ≠ 0, is called a linear polynomial.
A Geometric Example
If the side of a square is x, its perimeter is 4x. Since the highest power of x is 1, 4x is a linear polynomial.
| Side x (cm) | Perimeter 4x (cm) |
|---|---|
| 1 | 4 |
| 1.5 | 6 |
| 2 | 8 |
| 2.5 | 10 |
| 3 | 12 |
Notice what happens when the side increases by the same amount, 0.5 cm. The perimeter increases by the same amount each time, 2 cm. This constant change is a characteristic feature of linear behaviour.
Fixed Amount Plus Changing Amount
Many linear polynomials have two parts: a fixed amount and an amount that depends on the variable. Suppose a chess club charges ₹200 to join and ₹50 for every match played. If m matches are played, the total amount is 200 + 50m.
| Matches m | Amount ₹ |
|---|---|
| 1 | 250 |
| 2 | 300 |
| 3 | 350 |
| 4 | 400 |
| 5 | 450 |
| m | 200 + 50m |
In 50m + 200, the coefficient 50 is the change for each additional match, while 200 is the fixed starting amount.
Problem
A player pays ₹750 in total. How many matches were played?
- 1.The cost is 200 + 50m.
- 2.Set it equal to 750: 200 + 50m = 750.
- 3.Subtract 200: 50m = 550.
- 4.Divide by 50: m = 11.
- 5.So the player played 11 matches.
From Linear Polynomial to Linear Equation
A polynomial such as 2x + 10 is an expression. When we equate it to a number, such as 64, we obtain a linear equation. Solving that equation means finding the input value that produces the required output.
Problem
The sum of two numbers is 64, and one number is 10 more than the other. Find the numbers.
- 1.Let the smaller number be x.
- 2.Then the larger number is x + 10.
- 3.Their sum is 64, so x + (x + 10) = 64.
- 4.Simplify: 2x + 10 = 64.
- 5.Subtract 10: 2x = 54.
- 6.So x = 27.
- 7.The larger number is 37.
A Polynomial as an Input-Output Process
Another useful way to think about p(x) = 2x + 3 is as a machine. You put in a value of x, the rule multiplies it by 2 and adds 3, and the result is the output.
| Input x | Calculation | Output 2x + 3 |
|---|---|---|
| 4 | 2(4)+3 | 11 |
| −6 | 2(−6)+3 | −9 |
| 0 | 2(0)+3 | 3 |
| 5 | 2(5)+3 | 13 |
This input-output idea is the beginning of the idea of a function. For every allowed input, the rule gives a corresponding output. Here, 2x + 3 is a linear function because it comes from a linear polynomial.
Linear and Non-Linear Inputs-Outputs
The rectangle area 10x − x² can also be treated as an input-output rule, but it is not linear because x² appears. The output does not change by a constant amount when x increases by equal steps.
Practice Problems
- Evaluate 5x − 3 for x = 0, −1 and 2.
- Evaluate 7s² − 4s + 6 for s = 0, −3 and 4, and compare its behaviour with a linear polynomial.
- A mother is three times her child's present age. After 5 years, their ages add to 70. Find their present ages.
- Two positive integers differ by 63 and are in the ratio 2:5. Find them.
- A person has three times as many ₹2 coins as ₹5 coins. If the total value is ₹88, find the number of coins of each type.
- A 300-foot fence is cut into two pieces, with the longer piece four times the shorter. Find both lengths.
- The length of a rectangle is three more than twice its width and its perimeter is 24 cm. Find its dimensions.
Key Takeaways
• A linear polynomial has degree 1 and can be written as ax + b. • Equal changes in input produce equal changes in output. • The coefficient of the variable controls the per-unit change. • Equating a linear polynomial to a constant creates a linear equation. • A linear polynomial can be understood as an input-output rule.
Next, we study visual and numerical patterns whose successive values change by a constant amount.