Class 9 Maths Chapter 2: Introduction to Linear Polynomials — Notes, Examples & Practice

A polynomial in $x$ is a sum of terms of the form (number) × $x^{n}$, where every power $n$ is a whole number (0, 1, 2, 3, …). For example, $4x^3 - 2x + 7$ is a polynomial with three terms.

In each term the number in front is the coefficient; a term with no $x$ (like 7) is the constant term. The highest power of $x$ that appears with a non-zero coefficient is the degree.

Expressions such as $\dfrac{1}{x}$, $\sqrt{x}$ or $x^{-2}$ are not polynomials because the power of $x$ is not a whole number. By degree, polynomials are named constant (0), linear (1), quadratic (2) and cubic (3).

A linear polynomial $p(x) = ax + b$ ($a \neq 0$) changes by exactly $a$ every time $x$ increases by 1. To evaluate it, substitute a number: if $p(x) = 3x - 6$, then $p(4) = 3(4) - 6 = 6$.

A zero of a polynomial is a value of $x$ that makes it equal to 0. For $ax + b$, solve $ax + b = 0$ to get $x = -\dfrac{b}{a}$. Every linear polynomial has exactly one zero.

On a graph of $y = ax + b$, the zero is where the line crosses the x-axis, because there $y = 0$.

A pattern is linear when the difference between consecutive terms is constant. Look at matchstick squares in a row: 4, 7, 10, 13, … sticks. Each new square adds 3 sticks.

To write the rule, the constant difference becomes the coefficient of $n$. Then fix the constant by matching the first term: $3n + b$ must equal 4 when $n = 1$, so $b = 1$. Rule: $3n + 1$.

Making a table of $n$ and the value, and writing the differences underneath, is the fastest way to decide whether a pattern is linear.

Linear growth means a quantity increases by the same fixed amount in each unit of time: $V = (\text{start}) + (\text{rate}) \times t$, with a positive rate.

Linear decay means it decreases by the same fixed amount each unit: the rate is negative. A scooter bought for ₹80,000 that loses ₹6,000 of value every year has value $V = 80000 - 6000t$.

Not every change is linear. If something grows by the same percentage each year (like compound interest or bacteria doubling), the amount added keeps changing — that is not linear. Real models also have limits: the scooter's value cannot go below zero.

A relationship $y = ax + b$ links two quantities so that $y$ changes by $a$ whenever $x$ increases by 1. Plotting its pairs $(x, y)$ always gives a straight line.

The number $a$ controls the steepness and direction (the slope): $a > 0$ rises from left to right, $a < 0$ falls, and a bigger $|a|$ means a steeper line. The number $b$ is the y-intercept — where the line cuts the y-axis, since $x = 0$ gives $y = b$.

If you know two points of a linear relationship, you can find $a$ and $b$: $a = \dfrac{y_2 - y_1}{x_2 - x_1}$, then substitute one point to get $b$. Temperature conversion is a classic example.