2.1. Introduction

In Class IX, you have studied polynomials in one variable and their degrees. Recall that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of degree 1, 2y² – 3y + 4 is a polynomial in the variable y of degree 2, 5x³ – 4x² + x – 2 is a polynomial in the variable x of degree 3 and 7u⁶ – u⁴ + 4u² + u – 8 is a polynomial in the variable u of degree 6. Expressions like 1/x, x + 2, etc., are not polynomials.

A polynomial is a mathematical expression that contains variables raised to whole number powers. The degree is determined by the highest power of the variable present in the polynomial. For example, in the polynomial '5x³ – 4x² + x – 2', the highest power of 'x' is 3, thus the degree is 3. Polynomials can be classified based on their degree: degree 0 is a constant, degree 1 is linear, degree 2 is quadratic, etc. Expressions that involve negative or fractional powers of a variable do not qualify as polynomials.

Think of a polynomial like a recipe. Each term (like '4x', '2y²') is an ingredient, and the degree indicates how complex the dish is. A 'degree 1' dish is like a simple salad (just one layer of ingredients), while a 'degree 3' dish might be a lasagna, which has multiple layers and elements combined.

A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, 3x + 5, y + 2, x – 1, 3z + 4, u + 1, etc., are all linear polynomials. Polynomials such as 2x + 5 – x², x³ + 1, etc., are not linear polynomials. A polynomial of degree 2 is called a quadratic polynomial. The name ‘quadratic’ has been derived from the word ‘quadrate’, which means ‘square’. 2x² + 3x – 5, y² – 2, etc., are examples of quadratic polynomials.

Polynomials are categorized based on their degree into linear (degree 1), quadratic (degree 2), and cubic (degree 3). For example, a linear polynomial represents a straight line when graphed, while a quadratic polynomial forms a parabola. A linear polynomial has the form ax + b (where a is not zero), and a quadratic polynomial has the form ax² + bx + c (where a is not zero). Understanding these classifications helps in recognizing the shape of the polynomial graph.

Imagine the degrees of polynomials as different types of vehicles. Linear polynomials are like bicycles – simple and straight. Quadratic polynomials are more like motorcycles – they can take curves and get into more complex routes. Cubic polynomials are like cars with more functionalities – they offer diverse routes and experiences based on how they’re driven.

Now consider the polynomial p(x) = x² – 3x – 4. Then, putting x = 2 in the polynomial, we get p(2) = 2² – 3 × 2 – 4 = –6. The value ‘–6’, obtained by replacing x by 2 in x² – 3x – 4, is the value of x² – 3x – 4 at x = 2. Similarly, p(0) is the value of p(x) at x = 0, which is –4.

Evaluating a polynomial involves substituting a specific value for the variable and performing the operations as defined in the polynomial. For instance, with p(x) = x² – 3x – 4, if we want to find p(2), we replace every 'x' with '2' and calculate the result. This evaluation shows how polynomials can represent different values based on input.

Think of evaluating a polynomial like calculating expenses for a shopping trip based on the amount of products you purchase. Just as substituting different quantities into your expense formula gives different total costs, substituting various values into a polynomial changes the outcome.